So, two things (though it doesn’t sound like you’ve done much research into this trip, so I’d advise maybe giving it all a bit more thought?):
First, you don’t need to send your passport to the Thai embassy in London if you’re applying for a visa there. The UK is indeed part of the e-visa process, wherein you never hand over your passport but rather handle the whole thing electronically. More detail here:
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Second, and to correct or at least add additional info to a previous comment: While it can take 7 business days (or more!) for your Thailand Pass to be processed, that it will take that long isn’t actually a universal fact. Both of mine were approved in less than an hour — including one which was approved one minute after submitting it.