Hi Chris Mason,
A gain only arises when you dispose of an asset.
If your ex-partner was to transfer their share of the property to you then they will require to check if they have any gain to declare.
If they have lived in the property for the full time they have owned it and It is their only home then any gain would be covered by private residence relief.
If you sell the property in future and you have continued to live in the property as your only home for the full period of ownership, then your gain will also be covered by private residence relief.
Private Residence Relief (Self Assessment helpsheet HS283)
Thank you.