Definitely dig drainage as advised upthread.
After that it depends how much you like gardens. If it were me, I would give up on the lawn as it will always be a struggle to maintain, and concentrate on establishing plants and improving the soil.
Soil that is rich in organic matter is able to absorb a lot more moisture than the kind of knackered mix of compacted soil and rubble that building work leaves behind. It's very hard to improve the soil under a lawn but dead easy on planted beds and paths, because you just lay the new compost on top of the soil as a mulch. Keep doing it spring and autumn at first then annually if you can.
Lift the turf and turn it upside down, then cover it over with loads and loads of compost and manure). Make bark chip paths and plant moisture loving plants such as willow and astilbe. In really boggy areas you could even grow bog plants which will stabilise the soil and give you a bit of interest, and be wonderful for the pollinators, but I suspect with a bit of care you will be able to solve this problem.
Oh last thing, it may well be your roof guttering goes to a soakaway in your garden (ours does). If so, install water butts on your downpipes to catch rainwater. In summer you may well want this to water your newly established planting. In autumn, you can use your water butts as a buffer to spread out the effect of downpour rain storms. Draw off the collected water regularly and use it for flushing the loo, washing the car etc. so the water is being released gradually rather than falling all at once.