if you have just under 20K i would consider S2000, NA Supra MKIV, RX8, Integra Type R, Magna Ralliart, XR6, SV6. These are probably the best cars, in terms of performance, you can drive on your p's.
However you seem pretty keen on getting an R34 or R33. I would say get an R34 for roughly 12K, and with left over money, maybe make it look nice. The R34 will give you decent performance (assuming you dont expect it too beat the above mentioned cars at the traffic light grand prix), a nice looking car and (thanks too the p plate rules) a car that will hold its value...... so in 2/3 years time sell it and buy a turbo skyline/Silvia/JDM car.