It might be that you'd get a few more dukes in the Americas if the royal family had several generations with many sons who'd need titles. Otherwise, the title of duke seems to have been a fairly rare one to be given out by that point. Expect lots and lots of barons, baronets and the occasional viscount, not all of whom would be peers able to sit in the House of Lords. Higher titles, earl and above, seem mostly to be reserved for people who do truly great things, like the Duke of Wellington, and for royals, who tend to be given a tonne of lesser titles, presumably so they can be split off and given to younger sons over time.
As for the institution of the Crown, I expect it would be both closer and more distant. Closer, because I don't see Westminster accepting colonial parliaments as equally sovereign barring anything less than the collapse of the Empire. After all, that's what it took otl. Governors and Governors-General are likely to retain more power over their local parliament than the monarch would have over Westminster. They may continue for quite some time with the practice of the viceroy appointing members of the upper house, and sitting in on Cabinet meetings, though I expect it would eventually become convention that they don't. How long that would take would probably depend on the age of the colony and its stability. The FCO is hardly going to give more power to a colony that's in a shambles or perpetually on the verge of revolt. The relationship would also be more distant, as the monarch is unlikely to visit often, for security and admin reasons as much as anything, and therefore is unlikely to make anywhere near as many public appearences nor to have much sense of what the colonies are like on the ground beyond what their briefings tell them.
Probably the most obvious relationship would be symbolic. The monarch on the coins, statues and street names and the names of towns and Royal Societies and Prince/ss so-and-so Hospitals and the Speech from the Throne with all it's assorted traditions. Royal insignias on post boxes, and a tendancy for the upper class, the aspirational and the well educated to immitate the Oxbridge accent.
Also, expect far less people to have names like 'Earl' or 'Duke'. I'm fairly sure the ban on naming your children titles has been around a while.