And the idea of "the same pace" just doesn't make sense given the dynastic principle (which Rome lacked). Commanders and prime ministers and so on, sure - but not kings.
It is actually as easy to adapt the English system as the City-of-Rome system. I will take the USA as an example, since it is what I am familiar with, but it applies to other colonies/dominions just as well. Let's say the early kings/parliaments are committed to integrating the colonies into the British Empire over a 200 year period (1650-1850). Call it 6 million English, 3 million colonist, 1 million Scots, 1 million Irish in 1750.
So first lets take the House of Commons. Sure the vote will be restricted to not all men as OTL (less than 17% of the population), but we will need to equal out votes. The King and the English Parliament will want to spread the tax burden. So by 1750, the USA will have 17% of the House of Commons (1.5/8.5) and about half the tax burden of England. By 1850, the USA will be up to proportional representation based on population. And this shows the problem. To survive, the English Empire will need to become the true British empire. The weight of power migrates to America (USA and Canada) overtime.
House of Lords. There are about 1600 Lords today, and I will just use that number and the 1750 population. The King will need to appoint a lot of Lords to get even close to balance. Since England dominates the list, probably 1500 are English, so this implies 900 new Lords by 1850 or 450 by 1750. Or put another way 57 per colony. Ignoring for the moment the internal politics of England, this way higher than needed to keep the colonist loyal. So lets say you appoint 2 per decade per colony on average. More than enough to keep the colonist happy and they will never reach proportional representation.
Now there are at least two easy ways to do this using the traditional structures. Originally the Lords had duties of minimum number of men at arms, boats, etc. Just do the same thing in the USA. Just to pick numbers to illustrate, and I may have the ranks out of order. Any colonial Duke has to keep one regiment of redcoats at his cost or equivalent thing of value to the Empire such as ships, maintain infrastructure or annual gold payments. Marquis does 2 Battalions. Count is 1 battalion. Earl is 1 Company. Baron is one platoon. It does not a take any large leaps from what the King was doing in the 1300's, and using existing structures.
Or we take the existing colonial focused way. We basically copied the UK systems in 1780's with minor modifications (indirectly elected House of lord and King). And adding so many Dukes, Earls and Barons will cause so many issues with old families. And the UK wants tax revenue. There is an easy win/win. Trade seats in the House of Lords for tax revenue or military units freely useable for by the King/Prime Minister. So again, for say 300 seats in the House of Lords selected by the various colonial legislatures for lifetime appointments, the Colonies have to maintain a standing army of 10,000 and trained reserves of another 90,000 at the cost out of colonial tax revenue. George Washington becomes a regimental officer of some modest success in wars against France. Andrew Jackson is noted for fame with the "fighting Virginian" division at Waterloo. etc. The side benefit of this system beside fitting colonial mindset closer is it allows the King of England to appoint the Dukes and Baron in the normal fashion over time, who can also fit into House of Lords. The UK appoints one Duke about every 13 years on average, so it avoids the "unseemly" rush to appoint "undeserving Americans"
Note: In reality, it is probably easier to use a federation type mechanism to accomplish this item. If one crowns the King as the "Emperor of the Anglo-Saxon-Scots-Irish-Colonist Peoples", one can then form a new Imperial House of Commons and Imperial House of Lords/Senate. If one limits the upper house to a set number of delegates per area it works. For example, the most senior 100 English Lords sit on the Imperial House of Lords, the lesser ones sit on the old English/Irish/Scottish House of Lords. It sounds radical, it would be opposed by the powers that be in England, but it is less radical than what the Romans did. If you want to have an empire that last a thousand years, it helps to copy the Empire that lasted a thousand years, at least the wiser policies.
Now since England needs a navy, it makes sense to have the colonials raise the bulk of the British Army. I picked 100K, but by 1750 the USA could support 300K in a bind and by 1850 it will be millions. You could probably leave the Sea Lords and RN largely untouched except for some minimal number of officers that have to come from the colony each year. If you can have a "German Prince" running the RN at the start of WW1, you should be able to live with an "Bostonian Earl". It would make sense to move your Army training academy and GHQ to North American. West Point works fine with the Army GHQ located in NYC. If 90% of the time, the Land Lords come from outside of England, people can live with the Sea Lords being mostly from England.
Now we probably did enough to integrated the Monarchy/Nobility into the system, but it is easy to achieve a full King. We have Duke/Duchesses in the USA. If the Prince of Wales marries the Duchess of Toronto in 1815, we now have basically gone the full Roman integration. If every other King marries a Duchess from outside of England but in the Empire, everything works. And if it is critical to marry Princess, we simply declare the King of England Emperor and we can have plenty of King and Princes in the various colonies. So we now have the PoW marry the Princess of Toronto. By 1815, many American cities or state were far more powerful than the tiny areas in Germany that had "Kings"
And what I list is not the only way, but simply a way. Once the will exist to integrate the empire, the methods can be found. It was only an issue of the missing will and lack of understand of the fragility of an Empire that does not integrate its white colonies. There was a simple choice the English had. Have an English Empire that would last a couple hundred years or have a British Empire that has a real chance to last 1000. They chose the English Empire. People say India is too big for the British to keep, and this is true if we mean English. The British (USA/Canada/Aussies/NZ/South Africa/Ireland) is easily strong enough to rule India by force indefinitely. England could only afford a strong Navy and a very small army. The British could easily afford the RN and the world's most powerful Army.
IOTL, the modern American Empire is a successor state to the British Empire, we just use different forms to rule our Empire. We often just use old British bases or bases in old colonies. We gave up direct rule, but many countries have resemblances to the old British Empire. Just like the UK used to depose Sultans that cause too many issues, we do the same (Marcos, Sadaam, Qadafi, many Latin American leaders over the years.) Using the old British empire color the map Red philosophy, England probably should be painted blue and called an American Dominion. After all we told the UK they could not keep the Suez. We told them it was ok to get the Falklands back. This is similar to the UK telling the Aussies how much of the German Pacific colonies it could keep after WW1 or telling the 13 Colonies not to cross into the Mississippi river drainage.