Go go go! I'm working at the Games and have thoroughly checked this out...
First of all, their are medics, stewards and mobility assistance people (with nifty little golf buggies) everywhere so you'll be helped quickly and swiftly out if need be. They all have radios so can be with you in a flash.
Second of all, you have queue jumping rights from everything from toilets to water queues. You can't take water in by the way - airport security - I take a big empty bottle and fill it at the drinking fountain areas by every toilet. You can obviously also buy it!
Thirdly, I have been offered a seat on tubes, trains and buses every single time so far. It will get busier, but people are good - and the Olympic spirit kicks in too. And you. Can always pick up a baby on board badge from any tube station as back up to flash and commandeer a seat!
Fourth, if an emergency happens an ambulance will take you to the NEAREST hospital - so there won't be a mad dash across town. This is ok, you just might want to carry your notes with you.
Fifth, it's a once in a lifetime event. Even if something DOES happen, what a fab birth story you'll have!
Have fun!