HI Charlie
Kit rooms are a great stepping stone into the industry, I have worked in one for two years and it really helped, the learning curve in a kit room is really good, also the people you meet.
Most rental houses in the UK are based in London:
247KIT
Genesis Hire
Pro Motion
Aimimage
VMI
Alias Hire
Procam
Cameraworks
And there is a lot more , just google Broadcast Rental houses in London and you will find a lot, these days they are desperate for people.
Send out your CV, keep your Linkedin updated, and keep on trying.
All the best