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Hi Everyone,
My mum is in the UK on Certificate of Entitlement which she obtained through my dad who holds a British passport since birth. When, and is she able to apply for a British passport/naturalisation on this basis and what is the process, or does she have to keep renewing her COE? Is she allowed to travel freely with my dad to EU countries that do not require UK citizens to have a visa or does she have to apply for a schengen visa? Your advice is appreciated.

When did she marry your dad?

If she has the CoE-RoA because she is

a female Commonwealth citizen who is, or has been, married to a man with right of abode at any time before 31 Dec 1982.

Then she is probably not yet British.
If necessary, apply for a Schengen visa as the family member of an EU citizen, at least until Brexit is completed.
Or Naturalise?

Yes that is correct - they were married in 1974. So can she apply for naturalisation straight away? Thank you for your advice Vinny.

Yes, if she has been living in the UK, etc., and satisfies the requirements for naturalisation.

Thanks