As it's on a fused spur I guess it would be OK as the fuse will
protect the total of the shed's usage to 13A and therefore all
cables are protected.

The quick answer is yes. A fused spur can run multiple sockets as the total
load is limited to 13A by the fuse.

The slow answer is that just having a fused spur from the kitchen in a shed
might not be a good idea. It is much better to run proper armoured cable
from a separate circuit to outbuildings. Then it is possible to have better
discrimination, seperate RCD to the house, and more reliable earthing. It is
better to run a radial than a ring in this situation.

If you don't wish to do this, and I can see why you might not, ensure that
the kitchen ring is protected by a 30mA RCD. If it isn't, replace the FCU
with one that incorporates a 30mA RCD that will protect the outside wiring.
These are widely available. Ensure that you have an IP66 style socket on the
outside of your shed to plug the lawnmower into and you might just get away
regs wise without a kitchen ring RCD claiming that the kitchen sockets are
unlikely to be used for outside equipment when a more convenient dedicated
socket is provided for this express purpose. Better to replace your kitchen
MCB with an RCBO, though.

Christian.