Oh dear, that's annoying for you. Is your property a new build I wonder? It sounds like the soil has become badly compacted, which is especially a problem with clay soils. The panacea for all soil problems is to add organic matter (i.e. well rotted manure or compost) as this opens up the pores in the soil allowing the water to drain and also produces humic acid, which breaks down clay. This would need to be dup in very thoroughly and depending on the size of area effected, this could be a big job. You'd then have to re-level it all out and returf. As a half-way solution, which might solve the problem, you could go over your lawn with a garden fork and stick it with lots of holes and fill these holes in by brushing in horticultural sand to creat mini drainage channels. However, the tines on teh fork might not be able to penetrate deep enough to be much use, in which case some sort of drainage system might be the next step. In the meantime, do keep off your lawn completely while it's wet, otherwise you're just compacting it further and making the problem worse. Hopefully someone else can advise re garden drainage channels....

You may find a heavy spiked roller with hollow tines would help. Do the lawn area both ways, clear up the deposits and as Sid says, brush in grit. Dress the top with compost before you lay the turf.