Hydrangea flowers are best left on through the winter Alex, I never remove mine until the new leaves are fully formed in late spring, they look great if covered in frost in the winter......

Agree with Linclass, however has it gone brown due to being too dry, has it died off ? check for healthy wood, have not noticed them turning in for the winter just yet, but if yours is ok and healthy wood is present then follow Linclass advice, but dont prune it down as taking off all this years growth means no flowers next year, assuming it has died off, then plenty of water will be needed and you may see some activity from the base.

Thank you for the information.

The Hydrangea has been dead for at least a few weeks, but I was just wondering what to do while I remembered it.

I will leave the dead growth on and look out for the frost on it.

I will test the wood to make sure the actual plant hasn't actually died.