Hi Richie,
I know it's not quite what you asked, but the normal answer to "how fast can they go" is "that's the wrong question". The first objective is to have fun. A really common experience is for people to see cyclekarts going around a track and to think "hmm they're not very fast" and then they get to have a drive of one and they come off the track with an insane smile on their face.
That said, technically there's nothing stopping you designing and building a cyclekart that can do 70mph, but the safety risks will have gone up exponentially, the fun gone out of it etc etc.
Perhaps more usefully is the following info:
The aim is to keep the weight in the region of 100kg or less if you can (mine weighs about 120kg now)
The engine is pretty standard at about 6.5hp and torque of about 12Nm (you can play with the engine to increase the power if you want/need to)
Gearing of the chain drive is typically between 6:1 to 7:1 multiplied by the ratio range of the Torque Converter which is 2.7:1 to 0.9:1
Wheels and tyres give you a combined radius of 250-300mm.
Those factors plus the weight of the driver pretty much determine the performance until you start travelling fast and wind resistance kicks in.
My recommendation would be to build light (certainly lighter than mine!), go more towards the 7:1 end of the ratios and enjoy the fun. Oh and keep your weight as low as possible.
Marek is building a Delage 15S8 and is planning to use an electric powertrain so that's an alternative route. I don't think its been done in the UK but in the US people have found that something like a 3kW motor gives good performance. So, electric is also an option albeit less proven
Chris has answered about the number of meetings in the last year. I'll only add that the first meet was last September (2018) and things have developed fast in the last 12 months so I suspect we'll have even more going on over the next 12 months.
Re pre-war racing, there's plenty of places to get inspiration. Brooklands Museum is probably the home of this era and has plenty to see if you're ever nearby, but of course Google will show you 100's of options. How about an Aston Martin 'Razor' or an Aston Martin Halford Special...honestly, the list is endless.
If you haven't already found them, then watch some of Dennis Thomas's build videos on youTube.