I don't think this is a good idea OP
You'll pay about £75k in interest on a 20 year mortgage term, based on current rates, which are mega low just now, what if they go up?
You'll have solicitors fees and stamp duty to pay.
If it's not your main residence, then taking into consideration £175k profit, you'd be looking at potentially £30k(approx) in CGT.
How will house insurance work? I'm struggling to think of buildings insurance that will cover you for this kind of residential set up.
You will be liable for all maintenance costs, if your Mum stays in it for 20 years, it will need a new kitchen and bathroom during the course of that, if not during her time there, then at the end of it, whether you live in it or sell it.
When you factor all that in, it starts to erode the £175k fairly quickly.
If all was fair, the equity should be split 50/50 between you and your sister as at now, then anything subsequent to this over the next 20 years should be yours to keep, if the question of disposing of assets doesn't come into play, that is. (The fact DSis lived there longer than you is kind of irrelevant to the financials, though appreciate it is something you are factoring in on a personal level).
Lastly, heaven forbid, but what situation may arise if you, or your DH passes away?
Also, on a more positive note, what if your Mum gets a telegram from the reigning Monarch and becomes one of our country's centenarians?
Time for your Mum to have a reality check, if you still insist on trying to forge ahead then you need proper legal advice here, further to the mortgage broker, need to speak to a solicitor and an accountant.