If you have a Thai bank account, they can print out a statement with the address in English. If you're working, they will accept a letter from your employer confirming your address. Otherwise a Thai driving licence or a 90-day report receipt showing your address.
If you're in Thailand you apply through VFS in Bangkok or Chiang Mai. Depending on how busy HMPO are (in Liverpool) it can take anything from 3 to 10 weeks.
The most important requirement is that she can show a genuine reason to return to Thailand after the visit, like a job. You, as her sponsor, need to show you have suitable accommodation and finances to pay for the trip and her expenses while in the UK.
Kids are not fined or banned for overstaying so you can leave it for a few years. However his passport will be stamped 'overstay' which cause an issue if he wants a proper visa later.
It's happened to me a couple of times. I've always sat it out and always got the Approval, even after the due date. If you're worried, and it's convenient, go to Immigration, but the worst that can happen is you're fined 2,000 baht next time you have to go there. I bet it comes!