I'm using Three rather than EE, but L2TP is independent of the broadband connection (4G or otherwise) that you use. It is an outgoing VPN, which will work over a 4G connection, which basically provides a tunnel for outgoing and incoming traffic.
There are some guides at https://support.aa.net.uk/Category:Incoming_L2TP
Ideally you'd set your 4G router to always run the L2TP session so that your whole network benefits, but you can do it from a single computer if you prefer (or to test it out if without having to buy a suitable router).
Actually I've just checked, and the Nighthawk should do everything you need. Just plug the EE modem into the WAN port of the Nighthawk, and then use the OpenVPN feature of the Nighthawk to set up the L2TP VPN. (it is easier said than done, but that's the overall picture...)
The AAISP L2TP can be cancelled at any time, so no long contracts, and at £10 per month it won't cost much to test it out. Try just getting what you have working with the Nighthawk first, then go for the L2TP.
Edited by sheephouse (Thu 04-Oct-18 16:22:33)