Great idea! Me and friend from uni (Pete) made one a couple of years back, we used a lorry turbo of some sort from the scrap yard. We used a saab PAS pump for the oil supply and a wiper motor to drive it slowly enough not to over-pressurise the turbo. A spark plug and a relay wired in to itself to generate a constant on-off-on-off for a coil and spark (just to start the engine) (the relay didn't last long doing that

) We drilled out a regulator and ran the engine from a patio gas bottle and an electric leaf blower to start the engine up. The whole lot cost about £40 all from the scrappo

It can certainly be done cheap.
We had a proper Whittle moment and it was perched on a chair in the garden while my friend kept turning up the gas regulator and nothing happened, then it exploded into life and started accelerating (with the regulator fully open). We panicked and ran away as it accelerated like mad, then my friend grew a pair and ran back to pull the gas regulator off the bottle. It was so much fun!
This was before Furze had done any jet engine stuff. Colin Furze is a friend of my friend Pete. We did the engine in a weekend the other side of the country in Stamford, I went home and Pete showed Colin some time later, then Colin put the engine in a BBQ frame and made the worlds first Jet BBQ, it was pretty funny. We put a sausage in the exhaust heat and it got fired across the garden. I think one of the sky channels did a piece on the Jet BBQ. Colin is a very talented fabricator, and Pete is a very talented engineering, between them they've done some other jet powered things, The pulse jet bike being a good un. Colin has made a Jettle (that's right a jet powered Kettle), and Pete is currently working on a valve type pulse jet powered lawn mower. This is truely insane with 80% the power of the buzzing bomb!!