Digging a hole for themselves
So they have now stated to the court that an agreement was reached and now stated to the court that they did indeed make that agreement without reservation, albeit claiming they made a mistake in doing so. The letter to the court saying agreement had been reached must have been another mistake as well.
Their letter makes the facts clear - they offered to settle and I accepted. So I have now asked the court to change my claim to one for breach of contract whereby they have simply not paid the £30 they agreed to pay, and the letters to the court are my evidence of their unconditional offer and their agreement (telling the court we had reached agreement).
This will be a way simpler case to win - show the judge the letter where they offered, my reply accepting, (contract formed) their letter telling the court we agreed (proof they got my acceptance), and bingo.
Actually, thinking about it, even their claimed mistake does not change things. If their offer included "Subject to agreement", the fact they then wrote to the court saying an agreement had been reached means, well, that agreement had been reached and so the "subject to" is satisfied.