The problems you describe are typical of lawns on heavy soil and in shade, unfortunately. For the worst affected part with the most shade, if its possible, consider removing the grass from that area and turning it into a planted part instead. Regular spiking with a garden fork helps drainage a bit, and moss killer treatments can be used, but any areas where moss was prevalent and then is either raked out or dies will need to be reseeded. I recommend you get yourself a copy of D. G. Hessayon's The Lawn Expert, its not expensive, and tells you probably more than you ever wanted to know about lawns and how to keep them healthy.
Regarding waterlogging, it should be borne in mind that we've had much more rain in the last 18 months than we would normally get, so its possible you've seen worse problems with this than you could expect in other years.

Thanks for this - I've sent off for the book. Before it gets here can you give me any advice on the order I need to do things - scarify, applyweedkiller, sow seed - is that the correct order?

Most grass does recover if cared for, now your moss is removed what grass is left in those patches will start to grow as the weather gets warmer. I suggest those patches you do need to resow you do when there is active grass growth and perhaps choose a seed for shade which hopefully will survive better, but as already said its probably best to remove grass in those very shady areas
In the spring I rake mine (mower set on higher the first couple of mows) a few days after those first mows when there is some growth I feed and use a weed/mosskiller and repeat this after 6 weeks. Once growing I mow every 4-7 days keeping it short as regular mowing encourages side growth which thickens the lawn.
I too like the D. G. Hessayon books and found the lawn one helped a lot.

Thanks for that!

No, not quite the right order. If you want to reseed any areas, no granular lawn treatment applications should have been used for six weeks, particularly not those which include a weedkiller in the mix.
I always prefer to scarify in autumn rather than spring, but some people do it in Spring. Spiking is a good thing to do on heavy soils, and as for seeding, I'd do that now. Most lawn treatments start in April - but this year, its so cold, probably best left till May anyway. Even so, it should not be applied to the newly seeded areas for a good couple of months, so you either treat the lawn first, wait six weeks and seed the patches, or seed first and treat later.

Yes I agree but my order did not include seeding. I prefer to scarify in the growing months so that it recovers and looks ok over winter.

Drc, my response was in direct response to Bobo's second query about having the right order, and not intended as a response, comment or criticism of your answer. I didn't actually read your answer properly anyway, just skimmed it to see if you mentioned the problem with seed/lawn treatments.

Yes got it now Bamboo lol