Hi Antony, and welcome to the forum.
The K.I.S.S. theory is all well and good, but you have already made your first mistake, by reading through the forum!

I understand what you mean about building on a budget, I had to watch the pennies too, so to that end, I would lean towards a Mobility Scooter for the majority of the donor parts.
This will give you the Drive Train, Batteries (usually 2 x 35 a/h units), all the electrics, Soft start, Forward / Reverse, Electric Brake, and most of the Steering Components.
You should be able to pick up a reasonable (working) Scooter for around ?200 or less, from Ebay, but make sure EVERYTHING works properly. These things are a nightmare to fix if they are faulty, and very expensive to replace Controllers, so be careful.
If you use this method, you will get a couple of days continual usage from the batteries, before re-charging, giving more 'Play time' for the little ones.
Another advantage is the speed. If you find a 4 - 8 mph one, with careful gearing you should retain almost all of the top speed, and be able to keep it to walking pace at the flick of a switch.
As for the wheels, Try 'MSA Wheels and Castors' in Rochdale, they have an Ebay shop online, or 'Google' the name, and it will find it. This is where I bought the Trailer wheels for Harvey's car, under ?60 for 4 wheels and tyres.
One thing that is recommended with a Scooter drive system, is a 'Swinging Axle'. Because the back axle and body are so rigid, they tend to loose traction on uneven ground, due to one of the back wheels lifting. Not a major problem, but worth noting.
Hope this rambling answer has helped.
Keep us informed, and as Brian will tell you, The only stupid question, is the one you didn't ask!
Regards,
Mark.