Succession - How to approach

Parents employ one person to help on their land. Full time, they pay the PAYE using Basic Paye Tools to their business tax hmrc account. And also the nest pension contribution on the nest website. PAYE HMRC online account is in my fathers name. He has died. The professional advice was so long as my mother kept paying the employee and he kept accepting then that counted as his employment contract being continued with her. However while I still submit the correct tax using the Basic Paye Tools software I am very conscious that the employers name on all the accounts is still my fathers. I am his legal personal representative and executor during probate but I can see its still somewhat dubious that I am submitting all this with a deceased persons name on it... But the correct tax has been paid which would seem to be the main thing? I think what I need is a succession. Somehow to keep the employment and records the same but now with my mothers name on them. How might I approach this? I'm afraid I've let it drag on longer than I should. I'm a bit scared if I call the helpline as soon as I tell them my fathers dead they will cancel everything and I will have no way to pay the correct tax... Ideally they would just swap my fathers name and birth date on his account to my mothers. Or my mother would register a tax account and at the end of the tax year we would just start submitting neatly via her account. But that still would leave the question of how to get all the records into my mothers new account to ensure continuity? Any ideas on how this works would be appreciated, before I just have to call the helpline and throw myself on their mercy... Thanks

Hi,
If this is a sole trader scheme and the sole trader has passed away then a new scheme will be needed as the legal entity no longer exists.
You may also wish to research something called TUPE which allows the new scheme to treat the employees as having  continuous employment. Please contact the helpline on 0300 200 3200 for more advice on this. 
Thank you.