the bearings in those BB are held in a cage, its common for the cage to rust and disintegrate causing the cranks to flop about. you just need to replace that bearing cage and all will be good again.
specific tools you will most likely need - crank remover, a lock ring tool to get the lock ring off, often the tools that come with old bikes that everyone has kicking around somewhere will have the lock ring tool on there. big adjustable spanner to undo the cup. just clean it out and put back together.
i would use the tyre thats on there, doesnt look like theres anything at all wrong with it. even so 27 1x1/4 tyres are easy to get still, i bought some a couple weeks ago dirt cheap. fine for a budget racing bike like that if its just for rolling to the pub on etc premiercamping.com/27-x-1-1-4-road-bike-tyre/?catargetid=520002630000002196&cadevice=c&gclid=CO7vpvHF5r8CFQH3wgodjTYAqA
you could have it all done for under £10 and shouldnt take you long to sort out