Unfortunately all you can do is keep mowing the grass thereby cutting of the sprouting trees. The only alternative is to lift the turf and remove the roots.

I sympathise. I, too, have problems with trees from next door but the owner likes them where they are so there's nothing to be done but grin and bear it.

Well I would reveal the woody parts of the roots (which might mean lifting turf), drill into them making holes about quarter of an inch deep, then fill up the holes with SBK, a brushwood killer. It might have the unfortunate effect of killing the neighbour's tree eventually though...

You can legally prune the tree roots that are trespassing on your property but you cannot legally kill your neighbour's tree so prune with care and tell your neighbour what you're doing and why.
Regular mowing should remove the sproutings before they get to any sort of size. Depending on the tree you may also have seedlings from seed which has fallen - I live with a council sycamore tree outside my front door and the past few weeks I've been sweeping up the seeds every day (the tree gets pollarded tomorrow - hopefully a complete shave rather than just 'short back and sides'.

Whenever we've had this problem regular mowing has kept it under control.