YOu can have lots of mortgages at once if you earn a lot. So if roughly you can borrow 3 or 4x salary and the two mortgage added together are less than that you're fine. For most people they only earn 3 or 4x their first mortgage so they cannot get another. You say partner not spouse so I presume one of you has moved out but you were never married to each other.
One of you will need to buy the other out of the first place. If neither of you earns enough to support that mortgage after a buy out which prboably means increased mortgage then it will have to be sold in most cases if you aren't married, just cohabiting.