The common route is to buy the best kids 20" wheeled bike you can find and use the parts from that.
Fork and plastic headset. Fork is Raleigh 26 tpi threaded and 90mm OLN.
Not many 1inch threaded steerers around and not for 20 inch wheels - the steerer is long 237mm.
Choices seem to be: Keep headset, put up with existing fork and rebuild front wheel using existing hub and new Al rim. Find suspension fork from kids BMX, which would involve cutting down the head tube substantially and going with a longer stem - not keen on that. Find appropriate new fork or have one made.
Rather than cut the headtube (which I have seen done (online) rather successfully) a lot of people extend the steerer of the kids donor bike instead, I think that a general welder can do this. I seem to recall that part of a quill stem was used to ensure strength and perfect alignment. This will allow a standard quill (yours is an odd size from what I can gather) and a standard threaded headset.
Rear triangle. OLN is 113 mm. Keep SA hub or replace. Coldset to 120mm to increase options.
Yes cold set to use the geared alloy wheel from the kids bike.
Bottom Bracket. The shell measures a huge 78mm length. Raleigh 26tpi threads and the 'Fixed' cup is painted in at the factory. Options here are keep as is. Try to find a cotterless axle that will fit the existing cups and bearings. More drastic options cut down the BB shell and use a threadless BB system or have the threads re-tapped for 24 tpi
On eBay, I think it's bankrupt bike spares or whatever they're called, sell a 26tpi kit to convert your crank to a cotterless set up, it's what I was going to use until I found I had standard threading.
Paint. Powder coat, rattle can or respray
Armoutex powder coat in one of they're snazzy, metallic colours.
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Fantastic, love the thread title and the fact that you're inspired! 
