C'est très drôle, Marc.

I take it you've adjusted well to an Oz accent, then. Then again, need I remind you about one piece of funny news from a couple of years where someone on an Air New Zealand flight wished to go to Oakland (California) and ended up in Auckland (New Zealand) instead.
Anyway, to get back to the OP:
With a POD of 1776, have the Canadian people keep an accent that is noticeably more British than American.
Bonus points if the US remains as powerful and influential as OTL.
Well, the bonus points are easy - just basically change nothing about OTL. In Canada, though, it's somewhat easy. Historically Canada received lots of Scottish immigration; whilst most of it was concentrated in the Maritimes they were a decent enough present in both Upper and Lower Canada that it modified somewhat the accents of the UELs and American immigrants in general who migrated to Canada (indeed, the Canadian raising is in part an application of the Scottish Vowel Length Rule).
Now, note that originally North American English was shaped by two different groups of English speakers, who provided the founding dialects. Generally, North American English (including General American) was shaped by speakers of Southwest and Northern England; in New England, by contrast, the founding dialect of the Pilgrims and Puritans was from the East Anglia area, so some of the stereotypical characteristics of New England English can be traced directly to East Anglia (including non-rhoticity and certain values of the vowels. One value in particular that sets a traditional East Anglia accent off, and a Broad Norfolk accent in particular, is that in some cases where there is the GOAT vowel (RP /əʊ/, General American /oʊ/) an authentic Norfolk dialect may substitute with [ʊ] (the FOOT vowel). This got carried over to New England with the Pilgrims and Puritans, where it got lowered and centralized to [ɵ] (the vowel of the French "e caduc" or schwa). This sound largely died out in the 1940s in general usage, but some traces do persist - for example, I always can tell someone who's a local by the way they pronounce the name of Rhode Island. A true local would pronounce it as "rudd-DIE-lin" /rɵˈdaɪlɪn(d)/; an American from outside would pronounce it /ˌroʊd ˈaɪlɨnd/, which irritates me though I understand it.
So here's a possibility. Imagine that somehow Canada received as much heavy immigration from East Anglia early on as it did from Scotland. As such, the Canadas would have an accent similar to a traditional New England accent, mixed heavily with Scottish features. So the Canadian accent would be distinct and more "British". As in OTL, this would be enough to influence the accents of the UELs and any other immigrants from the United States - indeed, in TTL, it would be more so, and let's have much heavier immigration from New England than OTL just to reinforce the point. That would be our starting point. How does that sound?