My first kit never oscillated either. It's no surprise with the largely substandard components included with it. For example the 0.02 uF caps could measure anywhere from 0.01 to 0.04 uF.
For the second kit, I used my own polyester caps and transistors. Either 9018G or 2N3904 could be used as the local oscillator/ mixer. For V2 and V3 I used 2N3904s, which provide high gain similar to the 9018Hs that are in the circuit diagram, but not supplied with the kit.
For V4, either 2N3904 or a BC548B should work, paying attention to pinouts. I substituted this with a vintage Ferranti silicon diode...and it worked too
Germanium doides like OA70 don't work here unless the circuit is modified...I am yet to try this out myself.
For V5 to V7 I used S8050s. The driver and output transformers vary wildly in their resistance readings from kit to kit, but did work.
I've attached a photo showing all the five points that need to be bridged and the points that should be cut to allow additional components for the AF stage, as indicated.