Hello, this is Jim and I am a dual-qualified lawyer (UK and Republic of Ireland) and happy to help you today.
The law in terms of non-married couples states that you get out what you put in to the property. This is contrasted with being married where one spouse may have a claim despite paying nothing. So, your contributions to the mortgage balance and anything else of significance would mean you can expect some equity out of the property when it sells, or if you want to sell your share (you can sever the joint ownership to tenancy in common whereby you then have a distinct share to do what you want with).
3 valuations of the property would be needed and then you would need to work out the equity owed to yourself. You can apply to the county court to force a sale of the property if your ex doesn't agree, or doesn't agree to buy your share out. To get you off the mortgage it either needs to be sold or your ex remortgages the property.
You say you want to come off earlier than July - if your ex disagreed then unfortunately you are looking at several weeks if not months to force a sale. In all likelihood by the time the court lists a hearing for you, it would be past July, so you may as well agree to the remortgage but ensure that what you are offered is reasonable given 3 valuations of the property (take the average market price and work out your equity).
I hope this helps – if you can please give a positive rating by clicking 5 stars (at the top of your screen), I can answer any follow up questions at no extra charge and I will be credited for helping you today.