That's plenty of time to get your visa. London is usually pretty good.
Worst case scenario is you fly without the visa and hope it comes while you're traveling, or enter visa exempt and just use the visa next time. Unless you have a long history in Thailand this year, and you're worried you won't get in without the visa.
If that's the case, you can use a safe entry agent to get in this time, and then try the visa the next times. That is probably cheaper than changing a flight.
How do you not have documentation showing your location? You got a stamp in your passport when you entered Vietnam. Without that your visa is rejected because you are not proving you are in Vietnam.
Which embassy you choose is dependent on where you are. You must physically be inside of the country where that embassy has jurisdiction. For countries that have multiple embassies/consulates, their website will usually tell you what part of the country that consulate has jurisdiction over. This is from the US for example
Every Thai embassy in the world is on the e-visa system and only accepts applications through the online portal. You won't even be allowed onto the embassy property without the printed form that you receive after submitting your application online, which lets you go to the embassy to pay the fee for the visa.
You provide them your information, they send it to the immigration at the airport, and then when you get to the airport you send them a photo of yourself with what you are wearing, and an immigration officer will meet you and stamp you into Thailand visa exempt.
It's probably okay. But it will be at the discretion of the immigration officer. It only takes a few minutes to apply for a tourist visa online, so if you want to be safe, I would go ahead and do it.