If the baby or child doesn't have a visa, they will be stamped in visa exempt when they arrive at BKK. If they then overstay there is no fine or ban, but when they leave Thailand their passport will be stamped 'overstay'. This shouldn't be an issue, but if the parents plan to live in Thailand and if the child may eventually go to school in Thailand, I would get them an O visa as your dependent and then they can be added to your Extension of Stay.
Is there family/friends who can renew it in the UK for him? I believe it can be done online. Otherwise you might need to get proper advice from one of the specialist agents here.
Presumably your son is using a different passport to stay in Thailand? I don't think a UK passport can be renewed in Thailand without the expired passport. Your son can renew it in the UK when he goes back there.
Currently there are no restrictions between Bangkok and Pattaya. It's a popular weekend getaway for Bangkokians, as at least restaurants are open in Pattaya.
Kostja Grusnewitsch Just make sure you keep your receipt of posting (preferably EMS). I would allow up to 8 weeks before chasing it up. They have a backlog as people are understandably not wanting to go there.