I appreciate your mum is stressing over this but its unclear whether the leasing company could confirm service history or not:
Although they may be limited by GDPR, I would contact BCA and enquire with them regarding the service history (leasing company should be bale to tell you what BCA it was). The reason I would get yourselves to do it and not the dealer is so you get the info from them BCA themselves. If they confirm there was no history then it strengthens your case.
As for Digital Records - Much harder to forge than a service book which can be easily made up. The best record is a copy of the service invoices in all honesty.
My mum phoned the leasing company who said all their information, including service history, was passed wit the vehicle to British Car Auctions at the end of the lease
Although they may be limited by GDPR, I would contact BCA and enquire with them regarding the service history (leasing company should be bale to tell you what BCA it was). The reason I would get yourselves to do it and not the dealer is so you get the info from them BCA themselves. If they confirm there was no history then it strengthens your case.
As for Digital Records - Much harder to forge than a service book which can be easily made up. The best record is a copy of the service invoices in all honesty.