I have alongside some normal square pin 13 amp sockets some 5 amp round
pin sockets, the 13 amp works as normal but the 5 amp is switched on
from a light switch located elswhere.

The round sockets were installed maybe late 90s in an 80s house by the
previous owners, I was advised by an electrician that this was an old
format of being able to switch on table lamps from a light switch rather
than the wall socket.

I have no idea which circuit they are on ie socket ring or lighting ring.

Now my question, I have recently purchased a 20 watt LED external
floodlight that comes with a 13 amp square plug, now I need to increase
the length of the cable and was looking to put a 5amp round plug on it
at the same time so I can use from the unused round pin socket.

Will this be safe to do so? With regard to Amps/wattage etc

Secondly I may purchase another 10 watt floodlamp so would it be ok to
run both from that 5 amp socket?