Women's entitlement to vote began with the Representation of the People Act 1918. That allowed women over 30 years old, with certain other qualifications such as property ownership, to vote in the General Election.
A PDF of the Act is available at legislation.gov.uk. It's lengthy, and seems slightly vague on exactly how someone would end up on the register. I have not read the whole Act, but skimmed through sections that appeared relevant, and I did not find anything requiring documentary proof for entry onto the register.
From page 38 of the PDF, which is the page numbered 290 on the original:
- It shall be the duty of the registration officer to cause a house to house or other sufficient inquiry to be made, and to prepare ... lists ... of all persons appearing to be entitled to be registered
So the registration officer's team (the 'overseers') are instructed to tour their wards finding out who was entitled to vote. The Act does not instruct them to seek any particular evidence of entitlement.
On the following page is a section dealing with "Claims to be Registered", for those who were missed from the register or had erroneous entries, and this looks to be more informative:
- Any person who claims to be entitled to be registered ... may claim to be registered, or to be registered correctly, by sending to the registration officer a claim in the prescribed form
- The form of claim for a person making a claim on his own behalf shall contain a declaration of the qualification of the claimant to be registered, including a declaration that the claimant has attained the required age, and is a British subject
So it looks like a person's word was generally considered enough to be put on the electoral register. No birth certificate or other documents were required.
There is a subsequent section on Objections, by which any elector could object to an entry on the register. This provided a mechanism by which false claims of entitlement could be dealt with. And the officer or overseer's judgement would presumably apply, if they had reason to doubt a claim, but this is not made explicit in the text sections that I read.