Not sure if it is different in England, but I can't think why it would be - but here in Scotland, the answer is no. You are not required to pay a deposit unless you are paying for a new build off plan and it is then basically a reservation fee.
What happens is that the solicitors will ask for evidence of ability to pay from the other parties solicitors - this is usually a confirmation of mortgage letter together with information on any other funds needed to make up the total sale price - e.g. from savings, or from proceeds of another property. The bank/building society may choose to pay in their share of the purchase money the day before completion, but both the mortgage money and any other funds are only required to be with the solicitor in cleared funds on the sale date. That way, he has the funds available to pay for your new property, or to hand on to you. If you have a property purchase pending then he may request to hold the money until that goes through, he will have to provide evidence of your ability to pay to the other parties solictor from who you are buying.
Usually when you are buying and selling and using a mortgage to part pay, you never see any of the money unless of course you are down sizing or reducing in property value and you will get the remainder back.
I would find some other way to pay your rent deposit.