I reckon that all you can do for the moment is wait - and hope we get some drier and warmer weather soon. Try to keep off it if possible, and when it's drier, try to improve the drainage by spiking it all with a fork.

I have a few muddy 'bald' areas, and I shall re-seed them later on once things improve.

I'd bet that by mid summer you will be complaining about having to cut the grass so often and this mud patch will be full of long grass!
Nature's powers of recuperation are incredible.

Keep off it to limit the damage and wait.

I agree with Drc726. I have a patch in my garden that seems to get wetter in winter. I walk round it!!