The story you are being told is very odd.
Why would a cash buyer agree to rent it for a year and then pay the higher price? How would the solicitor do up this contract? Would it be binding on both parties?
It would not surprise me if the house comes back on the market.
Auctioneers tend to ignore bids which are contingent on the buyer selling their own house.
What you need to do is to sell your house first and then you will become a cash buyer. There is a small risk though that house prices will spike while you are looking for a house, but it's a risk worth taking.
When you go sale agreed on your own home, you can then put in proper offers on other houses.