There's nothing scary about it really. Don't check in until you get to the airport. They'll look in your passport for a visa. You haven't got one. Then show them the printed out ticket out of Thailand within the time frame allowed and they'll look at it, check you in for the flight, and you're set.
If it were me, I'd buy a cheap ticket out of Bangkok to say Kuala Lumpur on a low cost airline about 25 days after you arrive in Thailand. That will be enough to get them to let you board the flight in London, once you've done that, you're fine to follow the guidance offered above re extensions, border bounces etc
Don't do this 'check in stage' because you don't have a visa number. Check in at the airport. They will ask you for a visa OR a flight out of Thailand within the 45 days allowed for visa exempt (also, some airlines do not realise it has been changed from 30 days) so best bet is to buy a one-way throwaway ticket to somewhere like Malaysia so you can show them that when you check in. They need to see that you won't be their responsibilty if you get refused entry (you won't)