My Lao wife and British son crossed at Chong Mek but I had to arrange "hospital" transport to the ASQ in Bangkok. This was in August. Not cheap but the only way I could get them home.
The UK only issue ETDs (Emergency Travel Document) if you lose your passport and need to travel urgently. I would apply for the new one as quickly as possible, and you'll have to take the damaged one to Thai Immigration and see what they say.
I would. It took over 8 weeks to renew my son's passport. Once you submit the application it's sent to HMPO in Liverpool. Due to Christmas and Covid, I would think they are taking a while.