High frequency AM demodulation
I have access to basic resistors, capacitors, some transistors and
  diodes, other components would have to be sourced.


If you don't use the correct diode you will have problems. Take the 1N4001 diode. 

This one has a reverse recovery time of 2 us and given that your carrier is well over 1 MHz it just will not cut-the-mustard.



Picture source.

This one is specified at 30 us and will be useless.

So, check what diode you are using and if the specification in the data sheet does not show a figure then you can safely assume it will not work. Bridge rectifiers will not work and probably most of them will not specify reverse recovery time at all.

Alternatively, consider the 1N4148. These are really fast with a reverse recovery time of sub 10 ns and are ideal for what you want. Also consider the BAS16 if you are going surface mount. It is about 5 ns.