I'm not in any way basing my ATL on Albert or Victoria liking the Danes or their cause, on the contrary I assume that a greater German influence at the British Court/Government might have the Danish Government realise, that they should be more cautious towards S-H.
If this has the Danish Government abort the unilateral annexation of Slesvig, which anyway was a kind of panick reaction, we have a good starting point for a negotiated solution over S-H including all the consequences for Bismarck's later role. And basically all we need is Albert having the Danish Government see the British Government/Court as slightly more Pro-German.
Regards
Steffen Redbeard
But this 'influence' of German unity existed in OTL and the Danes saw no reason to be more cautious regarding the question. I just don't see how the Prince Consort surviving would make the Danish government take any different action regarding the Schleswig-Hosltein question. I believe, that whether Albert is dead or alive, the Danes are still going to unilaterally annex it. You would need POD a little further back to influence the Schleswig crisis IMO.
Albert's influence on the British government was negligible. Although during Peel's premiership his authority "beside the throne" so to speak was made apparent by the fact that he was present whenever Victoria met her ministers, or even met them alone, and had access to all her papers, he didn't really do anything OTL to suggest he had any great influence over the workings of the government. It was precisely because of his foreign origin that this happened -- he could not allow himself to be seen as a meddling consort.
And even as he was dying worked to defuse the Trent incident. And favored the abolition of slavery.
Albert surviving means Victoria does not isolate herself, avoiding the subsequent loss of popularity of the monarchy. It also means Victoria does not blame her oldest son for his father's death, which reduces the amount of estrangement between them, perhaps resulting Edward being allowed a more active role in government.
A live Albert would also have an influence through letters on his oldest daughter, Vicky, who shared his views but lacked his patience and diplomacy. This could lead to better relations between her and the the Prussian court as well as her and her son, the future Wilhelm II.
There's also the chance that Albert surviving means Queen Victoria has a 10th child. She might not be too old for another pregnancy.
The mention of a tenth child sparked my brain -- Albert surviving will definately effect the life of their youngest daughter, Beatrice. OTL, Beatrice stayed close to Victoria for the rest of her life, Victoria relying on her for almost everything. It even went so far for Victoria not to see her youngest daughter marry, although she allowed him to marry Henry of Battenberg, she had many possible suitors, such as Louis IV, the Grand Duke of Hesse, her sisters widower, and most importantly, the Prince Imperial, Napoléon (son of Napoléon III; thus Napoléon IV to some Bonapartists). Princess Beatrice marrying Napoléon IV would be quite interesting, even if he is in exile at this point... it means the survival of the line of Louis Bonaparte, who had been King of Holland.
We could also see a different husband for Princess Louise; Victoria desired to keep her home as well, and demanded her future husband reside in England, thus she ended up marrying the Marquis of Lorne.The son of William III was proposed, but Victoria was against it because he had a mistress and lived a dissolute life in Paris -- if Albert is alive, the Prince of Orange may have a different lifestyle (through butterflies), or Albert may simply see it as a good way to cement relations between the English monarchy and the Dutch one. Either way, we could see a possible continuation of the House of Orange in the male line -- and a continuing personal union between the Netherlands and Luxemburg.
The same goes for Princess Helena, who also stayed near her mother because she demanded it. Yet I'm unsure what foreign match could await her.