Nearly all housebuilders require buyers to exchange contracts on their new house within 28 days.
This means normally, you have to stomp up the 10% of the purchase price at that time.
This can, as you have found, cause problems if you need an exchange on your current home to fund part or all the deposit required to exchange contracts on your new home.
You have a buyer fro your existing home and that would appear to be progressing well.
I imagine that David Wilson Homes might have someone else who is in a position to buy your plot, and perhaps for a higher price?
Be aware that Barratt's financial year end is 30 June and I think the motivation for all of this is to get you to legally complete before then, even though your home may not be entirely finished or as good as it should be. My advice is to stipulate that you will not be legally completing until after Friday 7th July and see what their reaction is!
Not knowing what your contract states, it is impossible to advise except to say you should talk to your solicitor who hopefully isn't the one David Wilson Homes recommended or suggested solicitors, as that is the single most worse thing a new home buyer can do - use the builder's pet solicitor!
Regarding the completion date, this is more clear cut as housebuilders are required to adhere to requirement 3.2 of the Consumer Code for Home Builders which stipulates what dates should be given to buyers as certain stages of the home are completed.
If you did not get a copy of the Code, remind David Wilson Homes they have breached Requirement 1.2 already!