Remove the baseboard trim (skirting board??), and mill a grove for the flat or round wire behind it. The problem is that there's a danger of someone pounding a brad or staple into the baseboard trim, and hitting the cable.
It's really a pity that I can't find CAT5 with only two pairs of wires in the cable. It would be about 66% of the diameter of the common four pair CAT5.
You could also lift the carpet and route a shallow grove into the plywood subfloor for the flat CAT5 cable. Subfloors are 0.75" to
1.25" thick. 0.100 isn't going to weaken the flooring. Glue in place with hot melt glue. Fill the grove and cover with Fixall. Reinstall carpet with a carpet kicker. (I haven't tried this but it should work).