We are trying to sell our house and over the last month have started to get viewers so things are picking up. We have also seen our dream house and the estate agent rang us to see if we wanted to put a deposit on and sign a contract despite not having sold.
We have no savings and would have been taking the deposit out of our current house upon selling.
Would banks lend us money for the deposit as second time buyers, not having sold, or will we just have to wait till we sell?
Are you willing to take the risk of agreeing to buy the "dream house" when your own house has not sold? Unless your prospective vendors are willing to accept an offer from you that your purchase is dependent on the sale of your own house ( which they should not do as that might never happen), you should ( realistically) not commit to the purchase if you are dependent on your sale proceeds.
Otherwise, yes, someone maybe willing to lend you the money. But its a big risk.