I have a climbing hydrangea, quite happily growing on a north wall for 10 years, however, this summer, the leaves are all drooping and turning brown fast!..my husband says it's dead..is he correct, or is it still alive?

Something is wrong with it. Cut off a branchlet and check if the centre is greenish-white or brown. If it is brown, then the plant is dying.

It may also be waterlogged - check the drainage.