I know we've had a lot of rain, but it looks waterlogged. Might just be a trick of the photos though.
If it was my garden, I'd lose the "lawn" altogether and have more planting and a central paved, woodchipped, gravel area in the middle
I think you had your answer when you said the previous owners didn't have a lawn!It does appear to be heavily shaded which makes it trickier. If you're desperate for a lawn of some kind, you may have to raise the area. That still creates a problem if it's shady, as it'll never be a brilliant surface, but even moss is fine for kids to play on.
@mr197, you’re very welcome. Try not to over worry yourself about it. You will have a much better idea of what’s what when the Spring arrives. You can then see how the grass fares, if it’s still abit waterlogged, just (garden )fork it and put some garden sand down the holes. Failing all that, I think the idea of cutting the corner of grass out( we did that) and covering the soil with mulch, gravel, anything to your taste, or just leaving the soil bare and planting put some nice plants, is all a good plan.Us gardeners ( unfortunately!) have to exercise patience in new garden, to first , get a lay of the land, see where the sun travels during the day and plan out a new garden accordingly.Give it time.
As a few have said, I reckon leaving it until spring will be best. It's amazing how well grass can recver from damage. It may be best to get someone in to dig/lay a sump with a couple of drainage channels in the spring/summer.Were the ponds definitely removed, liners removed/totally shredded? We moved home with a 2 year old and one on the way. The first thing I did was dig them a good big pond to stare at and watch the bugs.