If I have downloaded BBC programmes from iPlayer on to my computer, will the different broadcast standards between the UK and US prevent their being successfully played on a US TV using HDMI?
If you're using the official BBC application to do this, yes, you can watch programs you've already downloaded even when you're overseas. See official guidance: http://iplayerhelp.external.bbc.co.uk/tv/watch_outside_uk
You cannot download any further programs, and the programs you've downloaded will only remain for 30 days. This is checked via IP. The idea presumably being that a UK resident (with a current TV license) can download shows and watch them while on holiday.
HDMI is just the method by which you connect your TV to your computer, what you see on the TV is a mirror of your computer screen, and there's no difference between UK and US TVs in that way. So if you have a laptop on which a program was downloaded, and you plug it directly into a TV with HDMI to watch, that should work.