It's always difficult when property values fall, especially to the extent where it dips below the outstanding loan amount and negative equity strikes. When there is no need to move, many borrowers will sit tight and try to ride out the storm. As long as the mortgage payment remains affordable, then the ups and downs of property prices have little bearing day to day.
It is when there is a need or strong desire to move that negative equity becomes a bigger problem. Selling the property will force the loss to be crystallised and any element of negative equity will need to be met from elsewhere. In this case, there are not adequate savings to pay off the outstanding mortgage and loan amounts, let alone put down a deposit on a new purchase.
You have taken the right step in contacting your current mortgage lender to see what options it can offer. It sounds like, in theory, you could take some or all of the mortgage to the new property in isolation but you would also need the permission of the secured loan provider for the whole borrowing to be carried across.
Whether it is likely to allow you to take the loan across and be secured against a property of lower value must be questionable. It could prove fruitless to market your property, only to see it all collapse if the secured loan provider refuses, so I think that you should get in touch with it now to see what it says.
There have been one or two lenders, such as Nationwide and Lloyds, that have sought to offer a negative equity mortgage option to existing borrowers who want to move house. Unfortunately, there are no mortgages aimed at those suffering negative equity in the open market and available to new borrowers. If your current lender is unable to help, then you will need to stay where you are and concentrate on reducing your debts.