I've been made redundant twice, both when my departments were closed leaving me with no job. First time was from a job I hated so wasn't unhappy to leave, second time was a job I really did like so it was a harder hit. Both times I immediately treated the job as old news and got on with my life, both times led me into far more interesting career options so they were actually external prods to go do something better.
As I see it, you have two options:
1. Make yourself bitter and nearly unemployable by spending your time appealing then suing for a job you're not going to get back. Even if you win, you'll get a small payout of a few thousand pounds but you'll have a public record of suing and if you're in a small industry then you're essentially going to self-blacklist. It's a nasty old world out there in employment and it's nowhere near as black/white as the unions would like you to think; even if you win, you're not getting your job back and could seriously hurt yourself in the long term.
2. Treat it as a kick up the bum to go do something else. Treat job hunting as a full-time job. If you're doing less than 6-7 hours a day job hunting then you're slacking and need to kick yourself. If you can't find job hunting tasks to occupy 6-7 hours a day then you don't know how to effectively job hunt and need help, speak to your union or one of the freebie job advisory services. Get yourself on a CV rewrite course somewhere, the JobCentre should be able to help with this.
On option 1, it's a particularly stupid public authority these days that sacks someone without building a bulletproof dossier and having legal opinion. When I sacked someone last year in my former role as Chairman of my local council's HR Committee, we had pages of legal opinion and validation of our approach at every warning stage up to final dismissal leaving absolutely no avenue for credible appeal or law suit. You also need to remember the role of a solicitor, their paid day-job is to sue, there's no money in it for them to say "Sorry Mr Bleep, you've no case and I can't help you", if they win they get paid, if they lose they get paid. If you really must sue then ask your union to pay the legal fees, if they won't do this then you have to ask yourself why not.