What happens if my partner moves out and no longer wants to be party to the equity loan agreement?

The Post Sales Help to Buy Agent will be able to arrange for a ‘Deed of Release’ which will release your partner from the obligation of having to repay the equity loan. Assuming your mortgage lender is content for this to take place and that you can provide evidence that you can meet your housing costs and still have a reasonable standard of living, permission should be a formality.