Hi, @lms77. It sounds like your mum's Samsung account still has her previous phone number associated with it. As long as she's in the UK, if your mum's password is not being accepted, she may wish to reset it via the following webpage:
Should this not facilitate her in gaining access to her account again, then we'd recommend that she contacts the Samsung Accounts Team on 0800 995 1079, as our colleagues will be able to assist her in getting her account details updated.
This will also help ensure that your mum can use the two-step verification as expected.
With regard to two-step verification for her Google account, that would need to be amended separately via Google.
Please let me know how you get on with this.