I missed out on both the £245/£285 folders and the £360 non-folder, but I was given a nice Voodoo Bizango and I have an old 26er full suspension MTB available.
What is the best setup that budget could buy, including a battery, for shopping/commuting up to 20 miles return, in fairly hilly terrain, mostly tarmac but a bit of steep rutted forest road too?
And what if I used an existing battery and still spent all the money?
The Bizango would have to be a hub motor because of chainstay shape.
Bizango is not the best donor because of the small chainwheel. When you have a motor helping you, your gearing is going to be too low.
Personally, I'd forget it, and get a bike with a normal sized chainwheel, then you can fit whatever kit you want and enjoy relaxed pedalling at whatever speed you go. My bike has 11T/48T gearing for top gear with 26" wheels, which I think is about right for a touring/commuting bike. If you have 29" wheels, that would be about 42T chainwheel size.
Depending on your weight, a 15A KT controller running at 48v should give the power and torque you need, and a 260 rpm 48v or 201 rpm 36v motor (run at 48v). AKM are the best motors IMHO because of the high internal reduction ratio and clutch between the hub and the motor disengages the internal gears, unlike Bafang, Shengyi, Mxus, etc. AKM 100 for a rider up to around 85kg and AKM 128 if above that weight. It's difficult to find 250w versions of the AKM128, but they normally have an easy-peal label. When the label falls off, it's the same as many other 250w motors, and the same as the 250w version, externally.
I prefer the freewheel versions of those motors because they're a bit wider and can handle more power, but then you're limited to 7 speeds, which is another reason to get a crappy 7-speed disc brake triangle frame catalogue bike as a donor, then make the improvements to it that you want, like decent forks, hollowtech crank, hydraulic brakes, handlebars, etc. If you're going to have a bike for ten years or more, you probably want to change all those things anyway, so you might as well start with something that just gives you the frame and a few other bits.
Good cheap 48v batteries can be got from Greenlance or Yose.
PSWPower do some good 48v kits with KT controllers when they have them in stock. That's probably the easiest and cheapest solution.
I wouldn't use a crank-motor on a touring/commuting bike. they generally don't have the reliability, and the ride is much less comfortable.
It's a shame you missed the £360 bike. That would have been perfect, and it's easy to upgrade if there were something about it that you wanted to improve.