Both parents, and the child, need to personally attend the Royal Thai Embassy in London. Thai citizenship by birth is passed on via the mother, so in your case the father will need to register the child as his.
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-Birth-&-Death-Registration.html
If you turn up in Thailand with a UK birth certificate hoping to arrange Thai citizenship for your baby you will most definitely come unstuck.
Call the embassy in London and get them to explain it to you.