But even cycling, you've still got the same problem - you want to travel (by bike, rather than walking), to forage (to save money), and possibly work your way around the UK, all at the same time. You can do one of those, and possibly two at once, but three will be impossible.
Yes, you can work your way through via temp agencies, but temping isn't steady, you wont always have the skills, etc (I know, I temped for a long time). And you can't give an address which is a tarp somewhere. Foraging takes time and skill, and even then, there might be a good chance you end up with next to nothing, especially during the winter. And where would you forage? Most land is privately owned, so you wont have permission to forage, or stay on it.
Getting a bike is a better idea, but I think the people who suggested doing part of a coastal path and see what is possible, what equipment is needed and just how practical it is are right. Anyone taking on such a trip need to a lot of planning and have the resources available behind them - expecting something to just turn up wont cut it. I'd assume for a start that the foraging part is marginal at best, so that means buying food locally, and that needs cash, which in turn means savings.