Which model of HP is it a deskjet, Photosmart or lazerjet?
ePrint is where you connect to the internet log onto your HP Connect account and send it to your printer via email it tell you in the instruction.
To print from the tablet you would need an app a small program allowing you to use your printer directly if it is connected via Wifi.
"What I had envisaged was that any device connected to my router, either by WiFi or by cable, would be able to print directly to the printer"
You say your printer it's connected to your PC via USB it is not connected to the router.
Is your printer wifi capable?
Have you switched the Wifi option on on the printer this is in the instruction manual. Instruction manual can be got from the HP site.
If it's Wifi capable you may have had the option to change from USB to Wifi when installing the programs or from the folder with HP programs.