Currently no lenders offering 100%+ (negative equity) mortgages in the UK. A couple of lenders do offer a facility where their existing clients can take advantage of a rate if they are in NE but as you already know Santander are not one of these lenders.

If you are on the Standard Variable Rate (SVR) which is the rate you pass onto after you finish your fixed rate or tracker rate, then you may be able to negotiate taking the mortgage on your own but quite often the lender will look for you to take on a whole new product to complete a transfer of equity but the problem is if there is no product for you to move onto then this cannot be done.

Have asked a colleague inside the Santander mortgage dept if you can switch to singla applicant while on the SVR, hopefully will post back later with an accurate answer.