USA was easy. Friendly staff at a local T-Mobile store recommended a $10 sim plus a $3/day package which gave unlimited calls, texts and data. Took less than 5 minutes to sign up and get online. No ID needed.
UK was more tricky. Bought 2 'data' sims of different sizes (standard & nano, GBP30 each) from vending machines in Heathrow airport, both before and after customs. The nano one (EE) never worked. The included adaptor was of low quality and with it the phone could never see the sim. Also it made it stick in the phone (Galaxy S3) - twice. I ended up cutting down the other sim (3) and that worked perfectly. Both offered a fixed amount of data for a month (2 & 3 GB) and once an account was opened online credit for texts and mobile network calls could be bought. Note however that 3 has 'adult content' filtering turned on by default and I found it impossible to disable. Even after switching it off and waiting the advised 24 hours (!) it still blocked sites. I got around it using my VPN service.
I called EE and they simply referred me to the vending machine company as I had no receipt (not offered by the vending machine). Usefully they also mocked me for paying that much for a sim (!). Got the same response from an EE shop.
On my return through Heathrow I got the contact info of the vending machine co (Monex Systems) and emailed them. Impressively they responded quickly with an apology and a full PayPal refund.
So take care when buying sims from the Heathrow machines. Read the
(confusing) pack descriptions very carefully. Best to select a pack that gives you the right (non-adapted) sim size for your device. You can get a cheaper deal elsewhere, obviously, but less convenience.