Hello CampervanLady!
Unfortunately, it is very difficult to say how long it will take to exchange. Your mortgage offer and searches are only part of the process. You have probably saved a couple of weeks by purchasing the searches direct from a previous purchaser, but there are a number of points which need to be dealt with before an exchange can take place.
Your solicitor is acting for your mortgage lender too and he has to study all the paperwork relating to the property. This will include looking at up to date title deeds to see whether there are any restrictions, covenants, consents and rights of way which need investigating.
The searches will show planning applications and your solicitor will want to see certificates, guarantees, building regulations etc. This is not being awkward - your lender will insist on paperwork being in place (or an appropriate indemnity policy)
I am assuming you are not buying a leasehold property. If you are, then there will be the landlord/management company to contact to provide answers to all of the leasehold enquiries - they will have to provide a leasehold pack (this takes weeks to provide).
As it's a new build, are there any communal areas which will be maintained by the management company, i.e. planting schemes or maintenance of parking areas? If so, your solicitor will need to study documents which set out costs to the purchaser, as well as maintenance schedules.
Has the new build reached a structurally complete stage? Or are you exchanging within 28 days but have a long stop completion date?
Hopefully your purchase is straight forward but unfortunately it is unknown until all evidence is gathered and enquiries raised and responded to.
Hope that helps!