If your current lender has paid for the valuation, you'll probably have to reimburse them for that. But if you do, you might be able to use the same valuation for your new lender.
Auctioneer won't mind as your choice of lender is irrelevant to them. Solicitor should be fine too as you haven't yet signed the original loan offer, so s/he's unlikely to have sent anything in to the original lender.
Original lender will probably be disappointed, but that's business. Go for it.