I would imagine everyone's lawn is damp & soggy after all the rain we've had these last few months. Check if there is a shade problem. If overhanging trees/shrubs & adjacent fences/buildings are the cause then they need to be considered. A shade lawn will always have moss. Trees & shrubs can be pruned but buildings cant.
Compaction of the existing lawn will also probably be a factor. That is worse if the underlying soil is clay based, but can be dealt with. Strenuous job though- you can hire a machine for a day rather than having to prod the whole area with a fork yourself.
There are lawn treatments which include a moss killing agent that are applied in the Spring. Check these out at a local GC/B&Q to see how to use & just as importantly when to apply.
The moss goes black & then needs scarifying to remove it. I think that you can hire some equipment to help you do that with such a large area.
If cost isnt too much of an issue, but time & energy are, then there are specific companies that do lawn maintenance. It apparently doesnt cost that much... but honestly not sure about that.
Removing your existing turf & getting it all prepared for reseeding, in April, is a heck of a lot of work! As is removing the existing turf & then getting the surface ok for laying new turf. J.