Hi Jacob,
I would suggest that those companies aren't really necessary, unless you have to get working the very moment you land. If you are happy to do standard hospitality style jobs (bar tending, cafe work, hotel room cleaning etc), you will probably be able to find something quite quickly. However, while employers will want you to be a flexible as possible in terms of shifts you work and being able to take whatever hours are available that week, be aware many will be suspicious if they suspect you will head off somewhere else after a few weeks, so never tell them you are on a travelling holiday unless it's already time limited work, such as at a festival or similar.
I think a good option is https://www.workaway.info/. I have used this myself in the past, and it's worth the money. It would cost £42 for you both to sign up for a year, and you can contact as many people as you want. Eseentially, it's people who want some odd jobs doing and they'll provide you with somewhere to stay in return. It can be gardening, hotel work, doing up an old house, bar work... anything you can think of someone will want you for it! It's a good way of seeing places you might not think to visit otherwise, although if you want to stay in big cities that's also an option.
The above poster is right in that there's a lots of places in the UK, despite it being very small, and to recommend places to visit without more information is tricky! However, I would say the top visited places are:
For City Life:
Manchester, Newcastle, Glasgow, Edinburgh, Birmingham
For a smaller town vibe (Even though some of them are cities!):
Brighton, Bristol, Oxford, Canterbury, Exeter, York, Durham
