Indeed. The calculations I've just posted indicate that buried 2-core 16mm² SWA would seemingly be OK for 13A of power, but not for lighting. To get acceptable voltage drop for lighting at 13A total load, the op would probably need 25mm². As I wrote yesterday, this means that a good few hundred pounds could be saved by having, say, (car) battery operated lighting.[Just to get back to the OP - Bob you can see that from the 'debate' there is some significant variance in the size of the cable needed to accommodate your desire to provide power to the shed.
Clearly your 3 core 1.5mm SWA is not okay and something in the range of 16/32mm cable would be required.
However, I agree that a generator would seem to be a (much cheaper) option to be seriously considered.
Kind Regards, John