Using mortage credit function for quick cash?

Just looking for some advice /thoughts from you lovely people (not been on here for a long time and normally just browse).


Currently have a 2 part lifetime tracker mortgage with 11 years left;


Part 1 (repayment): 42k @1.5%(BoE +0.75)
Part 2 (interest only): 62k @2.74%(BoE+1.99), currently overpaying by around £400-£500 per month to reduce the capital



We have just been gifted 55k from equity released when my mum downsized her house, and plan to use some /most of it for house improvements, but this may take a while. It's currently sitting in an instant access savings account earning 0.2%(!!!), as we really need to be able to access it, and in small amounts.


My question is simply would I be better using part of this as a lump sum to overpay Part 2 of our mortgage (unlimited overpayments)? My thinking is this would reduce the interest I am paying, but the credit facility could still be used to access the money.


Is this how it works, and is this a good idea? I would plan to continue to make the same monthly payments, regardless off the remaining capital.


Hope this makes sense