A duct from the ground to top floor would be your best bet, either
larger size PVC electrical conduit or a wastepipe of whatever size you
want.
I would make sure you have access to each end of this for future
upgrades.
The other way is to use the inside of cupboards or similar to run the
cat5's up inside, easy enough to get to in the future and out of the
way, that's the way I've done it.
..

SJW
A.C.S. Ltd.

Thanks for your reply. I totally agree a duct with access at both end is
the ideal solution. The problem is concealing it, I only have one
downstairs cupboard and that's under the stairs, unfortunately any holes
drilled in the top of the cupboard come out on the stairs. Maybe I can get
some slim but wide ducting that would not protrude the plaster.

You could cut a chase into the brickwork giving you more depth for
bigger conduit. The other option is buy a bending spring, then you can
route the conduit up to wherever you want. Just make sure it's not got
too many sharp bends, long smooth bends are better if possible. This
way you would have to run several smaller curved conduits rather than
one larger straight one.
...

SJW
A.C.S. Ltd.