The first people to contact are the lenders.
As things stand, your ex-GF and you have 'joint and several liability'. That means that, if the mortgage payments aren't met, the lender can pursue either (or both) of you for the money. So, if you fell behind with the payments, the lender can demand that your ex-GF pays every penny. (It's completely irrelevant that she no longer lives in the house, or has any interest in it. She has signed a contract guaranteeing that the payments will be met; the lender is not obliged to accept any changes to that contract).
Changing the mortgage to a single name puts the lender in a worse position than they're currently in, because they'd only have one person to pursue for debt if things went wrong. So you'll have to be able to convince the lender that you'll be able to meet the mortgage repayments from just one income, both now and in the future (when mortgage interest rates might rise considerably). i.e. you'll effectively be applying for a new mortgage, with the same assessment criteria that a new applicant would face.
The lender might refuse to accept your (joint) proposal, and insist that your ex-GF's name remains on the mortgage. (If so, there's absolutely nothing that you can do about it, other than to sell the house). If the lender agrees to you having a mortgage based upon your sole income, they will guide you through the legal process.
Chris