You can stay in Youth hostels which are inexpensive and geared up for walkers and cyclists www.yha.org.uk
Assuming you decide to walk there are a number of national trails which take you through the most beautiful areas www.nationaltrail.co.uk
Ordnance Survey Landranger maps are large scale maps covering the whole country. showing the landscape in great detail including minor roads and walking routes. Buy them from W.H. Smiths, Waterstones and other bookshops or www.ordnancesurvey.co.uk
£4 - £5,000 seems reasonable. Aim to start on May and finish in September/October thus doing your trip in the best months for weather. May sure you are physically fit and have all the correct equipment before you start like lightweight rainwear and in the case of walking, tough comfortable hiking boots, first aid kit etc. You may not actually be able to do the whole of the U.K. Plan ahead and select the most beautiful parts
Work out your budget - for food, lodging and
also running repairs incl clothes/shoes etc.
So per day - how will you feed yourself, even supermarket (bread/sandwhich meats etc) plus some vegtables (5 a day does not go away) - adds up.
And then even Youth Hostels usually charge a small fee (even those that still take 'help' as part of the costs). There will be the odd emergency stay in a B&B / IBIS/Travelodge - so 50 a night typical average (more in cities esp if you do not book ahead by a week or more).
then you have your extras
- repairs and maintainence of whatever 'free' transport and your clothes.
- Travel costs (bus - or train - annual/ period ticket ??) - even if it 1 time per week.
Add that up per day, and multiply by 360,, but I expect it to come to more than 10-15 per day (average).
Good luck with that one........ not possible you will have to either fly or take a ferry to explore the whole of the UK, only 3 of the UK countries are on the island of Britain.so you will not be walking/cycling on the Irish sea over to Northern Ireland UK.
in Scotland you should try Edinburgh and Glasgow and go to the highlands theres a brilllant cycle course . England you should try London is very expensive though , cornwalls a lovely city too . not to sure about wales or northern Ireland to be honest. the uk has a lot of lovely places to go, you should look at tour guides for England Scotland wales and northern Ireland.
Sounds more like a fantasy
It will take you years, are you going to live in a tent, and what about the winter.