I am* *sorry to hear this and hope you can come to some agreement (financially or otherwise) that suits you.
However, there are some lessons to be learned here for the future:
We started the process in June and the estate agent has hassled us, and our solicitor at least bi-weekly to get us completed by the end of August.
The Estate Agent has no part in deciding when your transaction will complete. Their role is to advertise properties and find buyers. Once they have done that, the actual legal work starts and the estate agent is no longer part of the actual conveyancing process. They can talk to all parties up and down the chain but any information they have is based on what someone has told them, but may not be the truth.
DH and I MOVED MOUNTAINS to make this so
You are just one cog in the wheel. If your solicitor has asked you to do something, then yes, do that as soon as possible. But if other cogs in the wheel don't do things quickly, then the transaction can only go as fast as the slowest cog.
Come end of August, we agree a completion date and exchange week before, we do a pre-exchange viewing and the next day our solicitor contacts their solicitor to exchange and we are told "they are not ready to exchange due to issues further up the chain".
Your solicitor won't know anything about a chain above your seller, but it is entirely possible that the Estate Agent did know, since they are acting for your sellers in the sale of their property. I definitely would be asking the EA why they didn't inform you as soon as the additional chain was added. You can't claim any compensation from the EA for this as they don't act for you. But you could renegotiate the purchase price to take into account the amount of financial loss this has caused you.
WHAT BLOODY CHAIN?!!! We have been misled about the existence of a chain. Apparently there are actually 5 in the chain and one is buying a new build which hasn't finished being built!
Since new builds notoriously overrun their expected build complete date, I would be pressurising the party buying the new build, to break the chain. New builds usually exchange on a "long-stop completion date", which means that the completion date is on notice and could be many months ahead. Not good for buyers below with their mortgages expiring.
we have given notice on our rental and will now need to move into short term rental, with 2 small DC, in the meantime, at great expense.
This is unfortunately on you. Solicitors advise never giving notice on a rental before exchange. Don't ever do this. Do not listen to EAs on this advice. You have left yourself in a pickle trying to sort another rental for an indeterminate time. Could you have a chat with your landlady about staying on for a while yet?
We have no idea when completion will now happen, and my husband is all up for backing out and moving into another rental.
Well, unless the party buying the new build splits the chain, so you can proceed, you will be waiting until the new build is ready to move into. Do you know at what stage the build is at currently? If you don't, ask the EA to find out. But take "with a pinch of salt" what they tell you.
I know that nothing is legally binding until exchange
Correct and another reason not to give notice on your rental until exchange has taken place. You could certainly renegotiate the purchase price because of being misled, but ein mind that it was your choice to hand in your notice when you hadn't exchanged, though the seller might be willing to reduce the price since they failed to disclose the additional chain.