The double socket makes it a little more complicated. Assuming the 1.5mm cable is installed by 'Method C' (clipped to a wall, buriied in plaster, loose inside a wall etc.) it has a current-carrying capacity of 20A - so, if there is a possibility that the downstream protection might be as high as 26A (13A fuses in each of two plugs), then 1.5mm² cable would not be adequate and you would, strictly speaking, need 2.5mm² cable.
In reality, given the loads you describe, there is no problem at all. However, if you wanted to be strictly within regulations, then there would be a number of options:
1...Change the double socket to a single one (and keep the 1.5mm² cable)
2...Install an FCU, with a 13A fuse, in the feed to the double socket (and keep the 1.5mm² cable)
3...Keep the double socket and change the cable to 2.5mm².
If you wanted more than one double socket to be fed by the spur, you would probably need 4mm² cable (at least for the feed to the first socket), but I don't think there is any question of your wanting that.
Kind Regards, John