Hi, American expat now living in UK here. OP, SenecaFalls has made some really good points (as is to be expected with that username): a lot of this depends on what kind of company and what part of the country. It also depends on whether you are there to represent the UK part of the company, or if you are interviewing for a job there (in that case go conservative and polished). I'd actually suggest you start by looking for someone in that office of the company's website at about an equivalent level to you or slightly above, and see what she is wearing in her photo.
IME, some of what's been described is true for, say, the big accounting/financial firms and much of BigLaw, especially in New York and somewhat in DC. Those are really their own world. But if you're in, say, advertising, marketing, and sales in many corporations, the clothes will be less boring/formal/90s Next and more colourful at the very least.
Also, Manhattan is totally its own glossy world. Someone working in a head office in Chicago or Atlanta or Houston would still be expected to be reasonably polished (clean neat hair, minimal makeup), but with local variations (for instance brighter colours in Atlanta or Houston). I probably would wear nude hose on the first day (if wearing a skirt suit or a dress), just to suss out what's expected, but you may well find they're not needed after. You don't have to have a big shiny manicure either - plenty of women just do short, clean, buffed or clear polish nails. And definitely no need to get a drastic course of teeth whitening, eek.
Finally, on the New-York-is-different thing, in large parts of the country and in many workplaces, it would be utterly gauche to ask questions about cost, etc of someone's engagement ring, especially of someone who is a colleague not a super close friend. Local customs vary lots.