From my own recent experience, and considering you have been approved for mortgage, a bank should be able to give you a bridging loan on up to 90% of the equity tied up in your existing property.
It might be best to get the bridging loan, and arrange a payback date with interest, of for example 4 months time. There will obviously be a few grand interest, but you have the added advantage of being able to search for the best offer for your property as opposed to being "up against the wall".
You could also stall the first payments on your new mortgage by a few months, so as to release pressure in the interim.
Get a written valuation from your EA, and approach the bank, with all the figures on the table, from there it's about agreeing the best rate on the bridging loan. We got 5.5%, if that's anything to go by.