... and will get bigger and bigger

with time.
but colloquially of course it will be known as simply "UK".
The constitutional structure will be a unitary, parliamentary, [non-codified] constitutional monarchy with local 'devolutions' of power, which shall be determined by Acts of the Parliament ( Westminster ) 'home country' by 'home country'. Thus even the old european archipelago UK 'home nations' of England, Scotland, Wales, Ireland will keep their parliaments ( named ''Assemblies" to distinguish from the only true Parliament ) with competence for strictly local matters, which in long run will extinguish most of the separatism arguments. The same mechanism for the overseas new 'home countries'.
https://en.wikipedia.org/wiki/Devolution_in_the_United_Kingdom
so integration will move into both directions - unification overally combined with devolution from within. Which shall cancel the inner contradictions and conditions for differing of interests in longer run.
The guaranteed participation of the local elites in both houses of the London Parliament ( incl. by creation of Peers outta the local big-shots ) would make thick and wide and solid demographic foundation of loyalists.
Thus even the "Americans" will be "Americans" without to be "English" or "British". The UK would transcend its british-ness in a seamless manner.
The Acts of Union could become a new territories as new 'home countries' accession serial mechanism into the UK. Quite similar to the USA expansion in the current timeline. This will suffice also the colonial loyalists expansionist appetites.
Visualize 1850es and Texas(es), California(s) etc. 'joining the Union", accending into UK.
The most developed territories will achieve and be granted with 'home country' status, while the rest will enjoy "crown dependancy" status. So, most probably initially in this timeline in 1770es there will be only the 4 european home nations plus 13-15 North American ones and few dozens of crown lands / directly administered territories ... For long time direct colonies without 'home country' status and integration into the UK shall remain also the sparsly populated territories. Like the extreme North, the deep tropical forrests ...
The 'non-settler' 'non-home country' territories later could also get 'devolved' as "Dominions" similarly to the Indian Empire of UK in OTL.
OTL France is not so good example because of the miniscule populations of the overseas constituences. ( OTL USA exceeded the 'metropoly' population by 1830es afair ).
--
The prohibition of slavery in OTL British Empire would create an analog of the USA OTL Civil War, too, but perhaps involving a deeper South (
https://en.wikipedia.org/wiki/Golden_Circle_(proposed_country) ) vs the rest of the UK.
If also in 1810es UK takes the souther cone of Southern America, then by the mid XXth century we might see UK-ization of all of the both Americas? Especially if massive European immigration is stimulated into the new territories to dilute the Spanish/Portugese layer there ... Then the denser and more massively populated iberianized territories of the Americas will evolve like Quebec, Louisiana and Porto Rico.