What do I need for internet connection?

[OldChap]

The piece of kit that I need connects a wired lan to mobile internet.

I could possibly do this via a laptop but to clarify there is no wifi

What is this piece of kit called?

[Charles Wirth]

Wireless router?

[OldChap]

The wan needs to be 3g or 4g mobile (cellular) with normal gigabit lan

[pirogue]

Except for the gigabit part, one of these should work: TL-MR3220, TL-MR3240.

http://www.tp-link.us/products/?categoryid=202

[Sparky]

Many mid/top tier routers come with USB ports for that kind of thing on them now.

I'd personally go with something like a nice upper tier Asus (RT-N66U for example) over a TP link. Their switches are fine in my experience but the routers, ehhhh not so keen on them.

[Otis11]

The wan needs to be 3g or 4g mobile (cellular) with normal gigabit lan

So they're called 4G/LTE Modems - They have USB ones that plug directly into your computer (or that can be shared over many normal routers with USB ports in the back), or full routers that have ethernet plugs. You can also take any laptop, create a wifi point with it, and then use USB connection to connect to phone/stick for internet. (Did that one week when my internet wasn't working to update all my BOINC WUs... No, I don't have issues...)