They are liable for the repossession costs if the mortgage was a joint mortgage. That's because under a joint mortgage, you are both "jointly and fully" liable.(posted 9182 days ago)Lenders chasing repossession shortfall don't usually go for full repossession costs - they are prepared to settle for considerably less than they say they are asking for provided you don't query their figures.
I know of no case where a lender has come after the next property... but I'm sure it's happened. They don't generally do damage to each other, though they can get a "charge" against your current property. You can usually expect specialist lenders to be more aggressive in this respect. Another good reason not to use them.
Lee