I read that the Thai Embassy in London are issuing CoE's for those who are married to a Thai national, either to a new Non-O visa applicant with supporting documentation or those already holding a valid extension based on marriage with a re-entry permit.

 

How about a Non-O for supporting a child in Thailand with supporting documentation or a valid extension for supporting a child in Thailand with  a re-entry permit? Does that qualify for a CoE at this time?

A non-o visa or extension of stay based upon being the parent of a Thai will be accepted to apply for a certificate of entry.

See: https://london.thaiembassy.org/en/publicservice/119247-requirements-for-certificate-of-entry-during-travel-restriction?page=5d6636cd15e39c3bd00072dd&menu=5f4b6eb3f6ae4b236972c562#4

^ Excellent, thanks.