My mum just brought me a wheelchair for my own use for the next two months and she got this from the red cross near the Business Design Centre in Angel, London.

She knew about it because she'd borrowed one from them before when her own mother (my elderly Spanish grandmother) came to the UK for my brother's wedding. She was here for a week.

Just give them a call and ask if they have one you can reserve for the dates your mother will be here.

They don't have so many of the self propelling chairs (the ones you push and steer with your hands), but it sounds like you'll be wanting the other kind that requires an assistant to push the occupant - which is what I have. (you might want some driving gloves for whoever will be doing most of the pushing).

I've only started using the wheelchair in the past few days, the tube is pretty much off limits, only a minority of stations can accommodate wheelchair users and the gap between platform and train is beyond tricky. Buses in London have been a pleasant surprise though. The wheelchair ramps work very well and if you have a handy bus route near you, it's just possible you might find it easier to use than a car.

The red cross require a deposit (can be cheque) I don't know how much my mum gave as a donation though.

EDIT TO ADD:
And don't forget to bring a surfeit of blankets and warm clothing!!!

I've been putting loads of layers on, dressing as warmly as I did for my recent trip to the Canadian countryside (thermals, fleeces, patagonia shells, hats, the works) and even in this relatively mild Spring weather we've been having in London it's been FREEZING!!!
Perhaps it's different if you're self propelling yourself in a wheelchair, but when you're being pushed you feel your temperature plummeting, no matter how many sweaters you've put on.

Definitely get a very warm rug for your mother and make sure she's wearing very thick socks!