The basic story is that you need to get the Thai birth certificate translated into English. After that, it seems like different European countries have different requirements. It seems like the UK is happy with just a translated birth certificate to issue a UK passport as long as you are listed as the father. The full/bigger story is to get the birth certificate legalized first at the Thai Ministry of Foreign Affairs and afterward at your Embassy. This is the recommended procedure as this way will give the best legal status. The Thai MFA has requirements for the translation into English, but you can have it done in the hall of the MFA in about 1-2 hours. You might very well do/need the same with your marriage certificate to prove that your baby is born in marriage.
90-days reporting and extensions are not related as such. When you entered Thailand back in February you got an "admitted-to-stay" date. This is actually your guide to the 90-days reporting. However, it seems a little strange that you have got your extension before the "admitted-to-stay" date. It could sound like they forgot to give you the first TM47 receipt together with your extension. However, your 90 days notification date starts when getting the extension.