See I think it'd be worth going to an independent mortgage broker (unless a person is so inclined to do the research themselves) to make sure you find not only a) a place that is willing to lend in your circumstances, but also b) the best deal that you can get.
The news about RBS is great, but I'd still definitely shop around!
Since the visa situation wasn't ever an issue with us (and now that I'm a UKC too, it's no longer a concern at all), my husband just does his own research by keeping his eyes on what are published as the most competitive rates out there... Well he even did his own spreadsheet with formulas & all that jazz (cos he's a geek!) with what the monthly payment would be on the interest rate available, duration, fees for switching, if there are extra charges for things such as solicitor costs or offers cashback...and then what that all works out to in terms of the net monthly payment.

I don't think he's done much with it for a year or more though, because we're on such a sweet deal right now (obtained pre-credit crunch) - we'd have to look long and hard to find anything better, and probably come up empty-handed on a better deal being offered right now.