[morzh]

22 AWG is plenty for EVERY SINGLE wire inside the radio, with a possible exception of filament winding bus wire if carrying currents in exess of 5A, which is only a case in a very few radios.
350V rating is fine too as the stabilized voltage at its highest point except right after rectifier rarely exceeds 300V.
The output of the transformer high voltage has its own wires, if have to replace them due to cracked insulation, use 600V wires though 350V is fine too as 600 or so volts output is the end to end and then you have double insulation as there are two wires, and towards the chassis it is half that.