I would suggest (as you say it's a new garden) you see what happens over this winter with the ex-pond.
You could probably just leave it in a puncture it a bit, but if it still stays really waterlogged you could either dig it out or create a 'bog garden', planting up with damp loving plants - could be quite nice and would be a haven for frogs (useful if you're thinking of a veg patch). I think there's some info on the Gardener's World webpage, but if you google it you'll find plenty of ideas.
With the wild onion bulbs, you may be fighting a long battle digging them up as they'll probably have spread quite alot. Can you wait till spring and see what they are, they might be really pretty.
The previous owner had a pond which has been filled in with good soil which I was going to grass over or turn into a veg plot. But the plastic pond liner has been left in. Do I have to dig it out first? DP thinks it will be enough to dig over the soil with a fork and stab the liner. But surely plastic doesn't break up or degrade.
The other thing I finding everywhere is, what I think are wild onions. The smell oniony and have a bulb on. Is this a case of more digging? Any natural ways to kill them off (can't use poisons with dogs and DC around). There are a couple of flower beds with some established plants in I'd like to keep. Would it help if put some kind of bark chipping down to prevent weeds and smelly onions coming back up?