The buyers hopefully should be quite flexible about the completion date as they are currently renting themselves.
Can you really find and move into a rented place in a couple of days? From what I remember about renting it seemed to take at least a week for the letting agent to follow up references, credit checks etc. Is there anyway of speeding this bit up e.g. taking along bank statements?
If we were to try and get somewhere before exchange is there any way you can guarantee that the landlord/agent won't rent the house out to someone else. From my understanding 'holding' deposits guarantee you nothing!