Honestly, it is a hard question to say whether or not it would better or worse. It would certainly change the progress of the New World as there is no Post-ARW Scenario that is still really a win that doesn't involve an institutional change regarding the Americas. Be it the appointment of a Governor-General of the American Mainland who has wide-ranging powers over an American Parliament or something else. Lets take that scenario - probably on the back of a great win for the British.
We've probably got some sort of constraints in terms of territory, but we also have people who are trying to settle further and further west.
We also have the policy of this American Mainland Authority determined by a combination of the Party who appointed them, the Party in power in Westminster, and the Parties in the American Parliament to some degree. How do they handle it? Does the Westminster party want them to restrain it? Accept it? Willing to fight on that grounds? (After all, you could have Westminster splits in terms of Pro-India, Pro-America, Pro-Everywhere, Pro-Isolation, etc).
I personally think that whilst you may see more willingness to partner with native groups (not nice, but more willingness) with the British than the Americans - since the British would appreciate having allies who have less political goals within the Empire. So the Iroquois may well become a major protectorate of the British in the New World, effectively creating a cultural divide between the Lakes, Coast and West of the Appalachians. They could also be weirdly interested in exploiting the railroad - It'd be a great twist to see the Iroquois work with British Financiers to build a railroad through their territory, collecting tolls on travellers through and using that to finance their own economy. (Whether the British let them keep it is another question ENTIRELY).
Equally - you could see the opposite, a brutal expansionist Governor given the leeway and popular support in the American Mainland to ride roughshod and make the Trail of Tears look like a comedy show in comparison.
In my opinion? I think the establishment of a more robust institutional pattern regarding overseas territories has the interesting twist of both leaving more Democracy (as to avoid another ARW, in the new territories and back in America), but also more local development. I think a pattern of regional Parliaments and Protectorates are a likely pattern after the ARW, much in the same way India was taken over via the Princely States and the Company Raj. This institutional improvement, and greater resources in both India and America leads me to believe that you're less likely to see the Opium Wars, but still have a richer British Empire, and one that is already a step away from a Federation itself.