I have been on the other side (of a more extreme version) of this.
On our completion date, we received no monies from our buyers and so were unable to complete the sale of our old property and purchase of the new one on that day, which was a Friday. Everything was packed ready to be moved and had to go into storage for almost a week whilst it was sorted and then we eventually moved. It meant our removals costs doubled and we had literally only the clothes we were standing in. Luckily we were able to keep the additional costs down by staying with family and pretty much just bought some cleans pants from the supermarket to keep us all going.
In our case there was a clear breach of contract and our buyers were found to be liable for the costs we therefore incurred. The issue actually was not with them or their solicitor, but with their buyers solicitor. However, they ignored all attempts to sort it in a reasonable and timely manner, didn't seem to understand that as we'd paid £££ for the legal contract to be drawn up we were going to enforce it through the courts if necessary as we weren't prepared to just write off nearly £2k. It did go all the way to court and the costs due to the contractual breach got passed all the way to the bottom of the chain who I think along with the legal costs ended up liable for a significant sum. I did feel sorry for them as ultimately (and it was never proven but only their assertion) I think their solicitors were at fault. I hope they pursued their solicitor and were reimbursed. However, they really shouldn't have just ignored the situation and hoped it would go away. As PP said upthread, you're not expected to just have £kk lying around, but equally nor is anyone else and others in the chain shouldn't be out of pocket because your solicitor didn't do their job properly- and clearly, neither should you.
I would suggest:
-check your home insurance, bank account, anything else that might have 'legal cover' through. We had it through our home insurance and they basically picked it up, sorted it for us and sent us a cheque when it was done. Hopefully you've got legal cover somewhere so you can have decent legal representation at no cost
- Check the contract of sale- this should state the time by which funds should be moved, if no time is stated I'm not sure they have a case to say there is breach of contract anyway
- Follow the complaints procedure for the firm you used and write to the people pursuing you to confirm you are doing so and ask for a reasonable amount of time to do this (few weeks maybe), obviously ensure that deadline works with whatever deadline you expect the complaint to be sorted
- If no response from your solicitor firm, escalate to their regulatory body
Good luck, it sounds like you weren't at fault at all so you shouldn't be liable (you may technically be liable as your name is on the contract that was breached (if indeed it was breached) but ultimately you should be able to pass that liability to your solicitor if they haven't executed their duties timely and effectively).