Op, why do you say Probate is quite an expensive process.
Has your mum nominated you as executor, or a solicitor? If only you, it is very straightforward, with minimal costs. I did mums last year, including shares. The actual probate form only needs an estimate of cost, so as long as you know name and type of shares (you could help your mum make a list now), then you can value via web. (Once you have probate, you can then send grant off to stockbrokers and get shares put in your name, if you are sole beneficiary. I had mums, DB wasn't interested, he had more cash)
We had no income tax to pay. Death tax (Presume you mean IHT) will have to be paid but can you estimate how much that may be - bearing in mind that your mum will have both your dads and her own allowance - and maybe she puts that into a joint account?
All other household costs should be frozen until probate is granted, eg utilities, water etc. Council tax is not due for a given time after Probate.
I paid for funeral deposit and sundries from own money, plus probate cost (200ish?), the bank gave me a cheque from her account for funeral invoice. That was all that needed paying up front.
Given that my mum had terminal cancer diagnosed a year before she died, I had already helped her sort finances, so I knew exactly what she had, where (most amalgamated into one institution) so doing the probate form was very easy, just needed house valuations, which we got after funeral. Would your mum let you help her do similar?
So the only sticking point I can see in your situation is paying IHT, if likely to be due, and paying for funeral deposit etc if you have no cash of your own. (My DB had none, doesn't work - not sure how he would have managed if I wasn't around, but presumably mum would have given him some money upfront)