I am aware this lady breeds several pairs of British Blues and also claims state welfare, so clearly she isn't a GCCF Registered Breeder otherwise would have stated this. It is a cash in hand cottage industry which she clearly is not declaring, she asked me to bring cash on collection.

You might want to consider the Tax Avoidance helpline? 0800 788 887

If the deal was 'cash-in-hand' I cannot see how the seller can now insist on you signing a cotract that was not signed at the time. However, the seller might try witholding the promised 'family history'.
The cash in hand status also could make it difficult if (& I hope this is not the case) the cat has any congenital/hereditary problems.