Buying at auction you typically have to pay a deposit up front (usually 10-15%) and then the balance within a set period, usually 4 weeks. You can fund with a mortgage (both residential and buy to let depending on your reason for purchase) and yes banks will lend. Mortgage processing times can be an issue, 4 weeks is pretty much a minimum (no idea why it takes so long) so it's essential to get an "agreement in principle" for the amount you think you'll need to borrow before the auction, get all your paperwork lined up, then schedule an appt to arrange the mortgage asap afterwards (ideally same/next day), then chase up the bank if necessary to keep things moving. The auction terms usually threaten penalties for failure to complete in time.
Another option is short term lending will be on offer at auction that can tide you over until arrange a mortgage but this will obvs cost you.