I turned up to get one for my wife, without her, and they initially refused. However I was referred to the Supervisor and pleaded and grovelled, with lots of wai-ing, and she relented.
Kids are not fined or banned for overstaying but their passport is stamped with an overstay stamp. If you're a tourist that should be of no concern. If you plan on staying here for a long time or if the kids will go to school here, I would keep them legal.
You have to include colour photocopies of every page of your old passport, including blank pages, when applying for a new UK passport from Bangkok. I don't think they will issue a new passport if you have lots of unused pages.
I would think the translated copy would be sufficient for a visa from London but someone more qualified than me, should confirm. I needed mine legalised and certified for an Extension applied for in Thailand.
Make sure your marriage certificate has been legalised in the country where you married and then certified by the Thai Embassy in that country. If the certificate is not in English it will also need to be translated.
if you're just staying here on holiday. However if you plan on a lengthy stay and if the child is likely to be educated here, I think it's better to remain legal.
As Biff says, there are no fines or bans for under-15s who overstay. They do get an overstay stamp in their passport though. If the baby will be living long-term in Thailand it might be better to legal but if it's short-term as you plan to settle in the UK (or Cambodia?) it's unlikely to be an issue.