Sorry this has happened to you; I can see that it has come as a shock, because you had expected a very different outcome. Under other financial circumstances, I'm sure the school would have liked to be saying something different too.
If you had been on a permanent contract, then at the two year point you might have had grounds to claim unfair dismissal if you met specific criteria. Before then, you could only claim it was unfair if it was discrimination against a protected characteristic. Unfair dismissal would (after 2 years) be if you were dismissed without due process being followed, in terms of a disciplinary or performance process, or without notice. Alternatively if the post itself were being made redundant, you might have been due some payout, but it would likely have been small. In all such cases, these days you have to pay yourself to take anything forward to tribunal and it's often deemed to be not worth the effort/cost.
As many PPs have said, fixed term contracts operate differently. It's treated as you being awarded a contract to deliver the work for a set period of time, with no automatic right of renewal at the end of the contract period. They are often used for posts that are funded, for example, by research grants; it protects the employer from having to keep the person on if the grant doesn't get renewed. I assume that the terms of your FTC have been met in the amount of notice of non-renewal they've given, if notice was required at all? If not, that might be breach of contract, but unfortunately that's the only thing I can think of that might not have been done "correctly".
(In response to PP: an HLTA teaches my child's class for two afternoons per week - I have no problem with this; they are working in partnership with the teacher as far as I'm concerned. The school is doing its best with severely strained resources. And the HLTA is excellent.)