We flew to Japan in October when my wife was 27 weeks pregnant. You need to have a doctors note confirming you're fit to fly from 28 weeks until 36 weeks as already mentioned. We got a note just in case and took all our paperwork with us.

We went HBO for the week (spent too much time reading these forums...) but had to go to the check in desk for the passport check - we were asked how far along she was and they were happy with the response, didn't ask to see the paperwork.

We were on an F award ticket so on board there were no issues, so if you've got the option of an upgrade I'd definitely take it!