which budget integrated amp Rotel, NAD or other

I just bought a Rotel RA-1520. (My dealer didn't have the 1062, which is, I think, a little less expensive. I'm not sure if the 1062 is still available or not.) I preferred the Rotel over the NAD amp I listened to. The Rotel sounded more detailed, less grainy, and seemed to handle transients better. The NAD didn't sound bad, however, and it was cheaper, but even my wife, whose ears aren't as experienced, remarked (unprompted) that the Rotel sounded better.
I haven't heard the Onkyo in question, but my father owned Onkyo gear in the past, and while it never sounded bad, I think Rotel or NAD would be better that than any Onkyo I've heard. I did audition an Integra (also made by Onkyo) at a different dealer, and found Rotel and NAD superior as well.
Again, different people will offer different opinions, so your mileage may vary!

Rega products are said to be nice, although I have no direct experience. The Rotel integrated amps do have a phono stage, which was another nice plus over NAD for me. I don't know about those Usher speakers (a dealer I visited did have a pair of Usher something-or-others hooked up to an Integra for demo purposes, but that's my sole experience with Ushers), but the 60 W Rotel RA-1520 was driving a pair of B&Ws in the dealer's, and B&W speakers are generally known for being wattage-happy. The dealer I bought from also said that the Rotel (also a similarly-spec'd NAD) should have no problem driving my very inefficient Smaller Advents. (I haven't hooked it up yet to test because my Advents are still in the shop.) So I don't think you Ushers should be an issue.