StepAway several things stick out from your post.
The vendors don't appear to know where the deeds to their house are and have contacted the solicitors they used when they bought the house to try to get a copy. It's just nonsense. They need to just order a copy from the bloody land registry.
If your solicitors have already raised enquiries then they do have the title deeds to the property. Title deeds form part of the draft contract papers that your solicitors receive from the sellers solicitors. They would have got them from Land Registry regardless because they need up to date ones. You can even obtain a copy yourself for £3. It may be that the solicitors have asked the vendors for a copy of the Lease (if there is one) because if they have the original it is better than obtaining a copy from Land Registry for an additional fee).
DH has texted the vendor and told him that are buyers are very unhappy and want to agree a completion date and exchange contracts ASAP. It's annoying because we have done everything to make the process smooth, including organising a mortgage that would be very quick (because the vendors said they want a quick sale) rather than the cheapest possible deal.
You don't get to decide when exchange takes place or how quick the sale will be and part of the answer as to why it is taking longer is in your post:
We are also waiting for the estate management company to respond to the standard questionnaire, and something about needing a copy of the original planning permission for the development. Management companies are notorious for their slow responses. They often want fees to provide certain information, which your seller has to pay for first.
It's probably no use moaning at your solicitor. They have asked for information/evidence and so far it has not been provided. Your solicitor cannot physically make the vendors solicitors respond any quicker nor does your solicitor have authority to approach the management company.
Find out the date when your solicitor received the draft contract papers - this is the date from when the process starts. It might be 4 weeks, but this will be date that your solicitor starts work on your file. Even so, whether it takes 4 weeks or 14 weeks, your solicitor is acting for you in a very expensive purchase - he has to ensure that legally you are protected in your purchase and you surely would want that too?