Somebody called into the legal hour on LBC last week, (you can call with legal questions and they usually have a QC or some hardcore lawyer who can answer just about anything) with pretty much this exact same problem.
From what I can remember the answer was pretty much along the lines of; with no contract, no terms are being broken, so you can’t claim damages for breach of contract when there is none. With no contract there’s also no agreed notice period, so by getting rid of you immediately or before the start date - they haven’t legally done anything wrong, or words to that effect.
I imagine it would be very difficult to get anything from them, even if you did - the cost of getting it (solicitors fees etc) would likely eclipse the amount you’d be awarded.
Give yourself a slap on the wrist and write it down as one of life’s silly mistakes.