Leejamie - I have been in exactly the same situation as you! I did take it to tribunal (as you still can - you're not out of time) but if my experience was anything to go by, just forget it and move on.
I had complete faith in the system, but was wholly let down by the final decision.
I was sacked for something that I didn't do, the tribunal agreed that I didn't do it, but ruled that on the information that the company had at the time, they had made a reasonable decision to sack me. The fact that the company would have come to a different conclusion had they have carried out a full and proper investigation seemed to pass them by.
On the upside, the tribunal did see through the lies that my ex-employers were telling, so it was quite satisfying seeing them unravel under cross examination.
On the whole though, it was a thoroughly unpleasant experience that dragged on for over a year, and I truly wouldn't recommend anyone to go through it. You've already got a new job (let me know how you managed that - I've found it so hard!) so just draw a line under it, get on with your life and keep in the forefront of your mind that whilst you are working for someone else you are merely a statistic who will never be fully appreciated!