Yes, it does depend on country. UK all online and many others. Still a number of countries don’t use it (eg Australia). You can check on the site if you can use the evisa system or not.
The Evisa website lists United Kingdom and British Overseas Territories so you should have applied to London in the first place. London is usually very quick with visas so you might be ok.
You can extend the Visa Exempt entry by another 30 days at an immigration office without a problem. You might have an issue at check-in with the airline insisting on a flight out of Thailand within 30 days. Some check and some don’t.
If they make a fuss you can say you intend to visit Singapore or somewhere else nearby and be prepared, if necessary, to book the flight there and then.