Cheapest way to travel from Köln to Bristol

Megabus does Cologne to Bristol.

£27 each way, takes 14 to 18 hours, change at London Victoria.

Tickets for December don't seem to be available yet.

NB: KLM will fly you via Amsterdam for £117 return, according to skyscanner, which looks like a better deal to me because it only takes 3.5 hours including the 1 change, so the round-trip difference between this and the coach journey is £63 for which you save 20+ hours travel, i.e. you pay £3 or so for each hour of travel time you save, so you'd do better financially by working even a minimum-wage job for the difference in travel time and then flying rather than taking the coach. (That's assuming that your travel time is non-productive).

Most likely by bus but you will need to make a booking and compare several different companies at the time you are ready to book. As usual 'cheapest' depends on a lot of things which makes it hard to give a detailed answer.

If you book three months (or more) in advance you can get much better prices for flights and for trains than when you are nearer the time of travel. The higher the demand for the travel date, the more likely you end up with high prices and even an hour earlier or later can make a lot of difference at times.

Prices can run from 15 pound one way on a low demand date, but I am afraid that during New Year 2016 will not be low demand. If you wait with booking till the day of travel you might find bottom level prices nearer 200 pound.