In the 'good old days' (if they ever existed), where every bank branch had a manager who could make discretionary decisions, you might have been able to convince a lender to grant you a loan based upon what you were due to receive in the near future from a will. However I'd guess that your chances of doing so when everything has to be approved by a computer are as close to nil as would make no difference. (Sorry!)
If you've applied for probate personally (rather than though a solicitor) the whole process should be really quick. (My father died in mid-October and I was driving a car, funded from his estate, before the end of November). If you've made the mistake of going through a solicitor (or if the relevant will named a solicitor as an executor - which is madness) then you need to hassle, hassle and hassle some more, to get them to do the job quickly.