Access to the full electoral register by the public is mandated by law. The current rules, which have been heavily amended since, are contained in
The Representation of the People (England and Wales) Regulations 2001 [SI 341/2001]. There are dozens of statutory instruments (11 last year alone) governing the work of registration officers so it is hardly surprising if their staff get a little confused over the exact rules governing the right of access. It is also interesting to note the wide number of individuals and organisations who are permitted to have copies of the full register.
Anyway to get back to the OP's problem, if he can get to the British Library, they hold the largest collection of old electoral registers, and when I last accessed them, many were on the open shelves, (although not those after 2001). By going down that route you may be able to do what AntonyMMM was able to do in 2020 (presumably pre pandemic!). That said, the day to day working of the BL is severely limited at present due to the effects of their massive data breach last year.
However since these registers will be paper copies it doesn't really help with finding someone once they move away from the known address. At best it might provide names of other people at the same address who might be NIcola's family (and so lead to a chance of finding her birth registration) or even a potential future husband etc.
Incidentally, an earlier posting mentioned using solicitors. They don't have access to the full register either but can make requests to the registration officers in specific cases. However it is far more likely they will employ an investigation agency, who will often use less than entirely legal methods to trace people (cf the various scandals involving the press and access to confidential personal information, aka phone hacking).
