For the sake of completeness I would like to add that flying might actually be the cheapest option for you.
While it sounds like an amazing adventure, walking takes a lot of time, and you still need to eat, at least sometimes you might want to sleep indoors, you might need to take a bus or a ferry every now and then. If I would have very little money, I would choose to hitch-hike, but I guess it is not for everyone.
Busses and trains could be cheaper for moderate distances, especially if there is a direct connection. But Georgia is not exactly close, and probably not that well connected to the UK. So you would need to change a few busses and trains on your way, which on its own could again be a great adventure, especially if you would stay longer in places. But maybe that's not what you are after.
Another option, which you haven't mentioned, would be using car sharing websites, such as blablacar. Some rides offered there are for pretty long distances. Could be worth checking it out if you absolutely want to go overland.
However it is hard to find any other means of public transport that would be as cheap as some of the flights offered. I have seen flights to Georgia for 30-50 Euros from places like Bukarest (Romania) or Vilnius (Lithuania). In case you do not have direct cheap flights from the UK, maybe you would be able to get to Lithuania, Romania or some other country with good connections cheap enough?
But yes, walk! Have an adventure of the lifetime! Only you really don't need to hire a boat, there already are ferries and a tunnel with a train.