Short answer would be, if the 13 colonies remained part of the UK, they would not be treated much differently then the UK possessions in Canada or Australia. They would eventually gain more and more sovereignty and finally become an independent state of the Commonwealth, but probably not by mid 19th century. More likely by the early to mid 20th century like Canada and Australia abovementioned.
However an America not gaining independence before 1800, may have been totally different from OTL's US even by 1850. So who knows what the situation might have looked like and what OTL queen Victoria might have decided to do with it if it became rebellious or malcontent at some point.
Some ideas to consider:
=> how far would British America have expanded from the original 13 colonies compared to OTL US? (Would Victoria pick a fight with Mexico over Texas and California? Or would it even take on the French Republic over Louisiana- fat chance of Napoleon selling it to them like he sold it to the fellow Republicans of the US.)
=> Would a British American Colonies still see the same influx of immigrants the US did? And how would this shape not only the historical makeup of the US, but also the dynamics in Europe? How would the Irish Potato Famine play out if 1/5 of the population had no place to start over in? Or what if instead they ended up in Mexico?
=> Would the UK still abolish slavery in all of its territories by 1820? And how would that play out in the Southern Colonies?
So just to make a point. If the 1776 revolution somehow didn't happen, today's US might be anything, from a patchwork of states like in the south of the continent, to a Mega-Canada rivaling India in size and population, to a Grand Mexico occupying everything south and west of Charleston... Even to a United States of Australia taking over the US's role OTL...
However an America not gaining independence before 1800, may have been totally different from OTL's US even by 1850. So who knows what the situation might have looked like and what OTL queen Victoria might have decided to do with it if it became rebellious or malcontent at some point.
Some ideas to consider:
=> how far would British America have expanded from the original 13 colonies compared to OTL US? (Would Victoria pick a fight with Mexico over Texas and California? Or would it even take on the French Republic over Louisiana- fat chance of Napoleon selling it to them like he sold it to the fellow Republicans of the US.)
=> Would a British American Colonies still see the same influx of immigrants the US did? And how would this shape not only the historical makeup of the US, but also the dynamics in Europe? How would the Irish Potato Famine play out if 1/5 of the population had no place to start over in? Or what if instead they ended up in Mexico?
=> Would the UK still abolish slavery in all of its territories by 1820? And how would that play out in the Southern Colonies?
So just to make a point. If the 1776 revolution somehow didn't happen, today's US might be anything, from a patchwork of states like in the south of the continent, to a Mega-Canada rivaling India in size and population, to a Grand Mexico occupying everything south and west of Charleston... Even to a United States of Australia taking over the US's role OTL...