Counter Proposal
I have to say, the point made earlier about Britain being much more interested in trading opportunities and partners than glorious conquest pretty much applies to all scenarios along these lines. As a result, it would be highly unlikely for Britain to develop any interest in conquering Japan. Rather, if an association between Britain and Japan is what you're interested in, I have a counter proposal to offer.
As it so happens, 1559 was both the year when when Elizabeth the First was crowned and Oda Nobunaga Eliminated the last remnants of opposition to his control of the Owari province. Now, this is a stretch and requires a dose of eccentric adventuring and wanderlust, but if say, a (fictitious) third son of a prominent noble decided to make his mark on the world by discovering new lands for the glory of the, quite lovely, Queen Elizabeth the first. As I've just implied, this venture would be a (doomed) attempt to gain the Virgin Queen's favor.
Anyway, let's say that this young aristocrat sets off early on in the spring of 1558 (yes, before the Queen's coronation) and, hearing rumors of the mysterious and exotic island of Zipang even further west than Cathay/China from the many sailors, traders and explorers he interviews and recruits for his mission, decides to aim for the most distant target available, for it could be none other than the best for the honour of Elizabeth I.
At this point, I propose that this young man (let's call him John Blackwood (I'm not sure about the name; I think it might belong to a Scottish Peer, but I'm not sure if I'm getting this from Sherlock Holmes or history)) successfully petitions the crown for patronage on his expedition (he has a clear plan, an experienced crew and Elizabeth is quite curious about the Far East). This patronage would take the form of a small flotilla of half a dozen ships, along with their crews, to augment his own as well as a selection of volunteer soldiers lured in with the promise of adventure and glory in distant lands.
Now ready and with Royal backing, Blackwood would set out from England with his flotilla and, after an extremely trying journey (he would lose 2 of his ships and 3/5 of his crew) lands in Owari in thee summer of 1559. There, he and his crew would make contact peaceful (or at least non-hostile) with some Nobunaga's retainers and succeed in communicating that they wished for an audience with their leader. Intrigued by the novelty of these strange foreigners, rather than reacting with hostility as man if Japan would, Nobunaga instead chooses to welcome these strange men (who he understands to be emissaries of some sort).
Now, before setting off on his expedition, John Blackwood had been a captain in the Royal Navy (I've decided it would be best if he were not quite as young as I implied; this sort of position would require him to be at least 30 something) and, as I'm sure you'll be aware, Royal Navy discipline was quite strict and, while Blackwood was a highly charismatic and charming man, he remained a Royal Navy officer. As such, he treated his expedition like a Navy command which lead to his crew being, unusually for both sailors and soldiers, quite a bit cleaner and more disciplined than was typical for European explorers while in foreign lands.
The discipline displayed by Blackwood's men (he had them drill every morning now that they no longer had the excuse of being on a ship) impressed Nobunaga, which lead the Daimyo to entertaining the notion that he could use these men. And so, driven by his need to seize an opportunity when he saw one, Nobunaga would make a good faith attempt at successfully communicating with Captain Blackwood.
Both parties involved being exceptionally intelligent men, this progressed somewhat more quickly than one might expect and, after about 2 months, the two had grasped enough of each others languages to be able to at least understand one another. Mind you, neither would be writing the next great work of literature in their newly learned language, but their mastery was sufficient for direct communication.
At this point, Nobunaga decided that the time had come to make his attempt at enlisting his very interesting guests. He summoned Captain Blackwood and revealed his ambition of uniting Japan under his banner before rather bluntly asking what he would need to do to acquire the services of the Captain and his men in his endeavor.
Here, Blackwood say his chance. He named his price for assisting Nobunaga in subjugating the rest of Japan to be that the Daimyo swear to and sign a treaty of perpetual fellowship and alliance between himself and his successors and his Queen, Elizabeth I. Thinking that committing to an alliance with a monarch more a year away even on the fastest of ships was not much of one, he agreed to Blackwoods terms and signed the document. For the sake of posterity, two copies were made: one in English and one in Japanese, with the signatures one each being in their respective alphabets (it took some time to romanize Nobunaga's name and Blackwood's name underwent a quite literal translation).
In any case, Nobunaga had his allies. Captain Blackwood and his men would prove to be as good as their word and provided a great deal of assistance to the warlord during his campaign for the subjugation of Japan; the 5 galleons (and, more importantly, the cannons they carried) were very useful indeed in pacifying coastal fortresses and softening up often tightly packed armies who venture to close to the coast. In fact, they proved so effective that they would soon make themselves almost obsolete as Nobunaga's foes, not being suicidal, soon began to avoid marching or making camp within sight of the coast.
Once this partial blockade had been established, the energetic Captain Blackwood soon began to feel restless and, along with his full complement of soldiers, offered his assistance in land battles to Nobunaga (which the ambitious Daimyo promptly accepted). As has been evoked above, the Captain held his men to a very high standard of discipline and, while not being numerous enough to really turn the tide in many battles directly, they acquitted themselves quite well and rather impressed several of the samurai with their ''steel''.
Now, it must be mentioned the particularly impressive combat discipline of Captain Blackwood's men, while indeed exceptional for the time, made a greater impression on the Japanese than it probably should as they were used to noble warriors of great skill and half trained peasants and not familiar with the professional soldiers such as these men (The Elizabethan period was the last where English Nobles maintained highly trained retinues of professional soldiers at all times and men such as these would soon become unusual in the British Isles).
Now here is the point I was driving towards with this scenario. The result of Captain Blackwood's men assisting Nobunaga's forces during his campaign meant that, instead of missionaries and merchants (the latter of which the Japanese were highly disdainful of), their first contact with westerners was with, in their own terms, noble warriors who fought alongside them.
Now this first impression of the English would most likely have been fleeting, were it not for the events that took place in 1568, during his campaign to install Ashikaga Yoshiaki, who was the brother of the previous shogun whom had been murdered, leading to the former's thirst for vengeance. Now Captain Blackwood held the typical nobleman's stance on regicide, and was quite enthusiastic in his support for this campaign, not being aware that Nobunaga intended to use Yoshiaki as a puppet.
Thus, later during the campaign, Nobunaga, along with the soon-too-be-puppet Ahikaga Yoshiaki, came to speak with the Captain while he was in the process of directing his men during the usual morning drills. These were executed some distance from the main host as they usually took up quite a bit of space and made a lot of noise, so the Daimyo and Shogun were quite isolated from their own forces. Thus, it is hardly a surprise that they would be ambushed. Now, Blackwood had just under 100 men with him in total, so it was immediately obvious to him that he would not be able to repulse the 600 man strong ambush they had fallen afoul of. Thus, since the man was a stereotype on legs, he ordered his second in command to ''tell Her Majesty I'll be right behind the rest of you and back before Christmas'' before ordering a small detachment of his men (lead by his adjutant) to escort Nobunaga and the Shogun back to their camp. After that, they were to return to England and deliver his report of his activities since his arrival (it had been a constant work in progress) to the Queen.
After speaking to his adjutant, he requested that Nobunaga and the soon-to-be Shogun ''join up with your men'' and that he would be ''right behind them''. Unsurprisingly, the two men agreed and the English escort successfully retreated with both principles intact. The last known words of Captain Blackwood are : ''Gentlemen, fix bayonets!''
After they had successfully reunited with Nobunaga's forces, Captain Blackwood's adjutant informed the Daimyo the he and his comrades would have to return to England in order to carry out the Captain's final orders. Seeing little to gain in forcing the issue, Nobunaga release them from their promise to assist with his war and allowed them leave to return to their homeland.
Now, this sequence of events might well have been without continuity, were it not for the fact that one of those who had been most impressed by the Captain and his men was Tokugawa Ieyasu.
Thus, when, in 1599, an English delegation representing the now quite old Elizabeth I (and more importantly, the Royal Navy escort that accompanied them) came to Japan to normalize relations between it and England, they were greeted with uncharacteristic enthusiasm by Tokugawa as he had only recently succeeded the late Toyotomi Hideyoshi and, remembering Captain Blackwood's men as worthy warriors, extended that distant impression to the new arrivals and wished to secure their assistance in solidifying and asserting his rule.
After the English envoys had verified that Tokugawa was indeed the successor of the Oda Nobunaga who had signed the treaty, they agreed to provide a measure of assistance, seeing it as a good way of securing favorable terms.
From then on, Tokugawas' completion of the unification of Japan proceeded largely as in OTL with the exception that his forces fared quite a bit better in terms of supplies as the Royal Navy contingent escorting the English delegation was able to ''secure'' a great deal of provisions from nearby islands who were not blessed with much in the way of formal structure at little cost to themselves.
In any case, the battle of Sekigahara goes as OTL and soon Tokugawa Ieyasu rules as Shogun over all of Japan. After completing the unification of the islands, Tokugawa agrees to discuss the specifics of the association between England and Japan. The head of the delegation, being quite observant and possessing a keen mind, had noticed the disdain with which the Japanese seemed to regard commerce and, more importantly, that said disdain was transferred fully to traders.
Reasoning that there were many other islands in the far east and that bringing up the subject of trade would likely accomplish little beyond alienating his host somewhat, he instead spoke of the Crown's interest in the Spice Islands (Indonesia/Malaysia/Philippines) and that it would be most grateful if the Shogun would assent to allowing the Royal Navy (and later East Indiamen) use of their ports as bases for their enterprises in the region as well as the opportunity to recruit idle Japanese youths for said enterprises whenever manpower should prove scarce.
In exchange, England committed itself to providing a small Royal Navy Fleet as well as a regiment of foot at all times to assist the Shogun in ''policing and safeguarding his realm'' as a ''gesture of goodwill and fellowship''. This sort of agreement proved quite amenable to Tokugawa as he saw it as a way of acquiring resources beyond the reach of any potential rivals while simultaneously allowing and outlet for the destructive potential of restless youths. Therefore, he confidently signed the agreement, unaware that,in doing so, he had change the fate of his realm forever.
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How about that for a silly fantasy, eh? I know it's ridiculous and implausible to the point that it might actually have been impossible, but I quite got carried away once I got started and this idea got away from me. I know it's lacking in hard numbers, but I found this to be an amusing diversion nonetheless.
So, on a scale of 1 to the Ministry of Silly Walks, how silly is this?