Lemonwords, how stressful. You need to get onto your solicitor and kick their arse. Big time. Check also that your insurance covers you. Do not be fobbed off by your solicitor. You need them to deal with this as a priority, and to establish exactly what has gone on.
Without knowing all of the facts, hard to comment. However, I am surprised that their non-ability to move was not flagged until now. Usually, money is transferred only when parties are able and willing to complete (and that means vacant possession unless something is expressly agreed to the contrary) and the keys are handed over. Like another poster said, it may be worth finding out where the money is. If it hasn't gone through yet, perhaps there is time to stop it. Are you absolutely sure completion has taken place? You really need to speak to your solicitor and find out exactly what is going on.
If you haven't actually completed, but they delay completion, then the contract usually provides for them to pay you compensation. (If they knew they couldn't move out by the completion date, then this is what I would have expected them to do - delay completion).
If however, you have actually completed and they have your money, then it's a different ball game. And your solicitor needs to play hard ball. (Escalate if you're not happy with the way he/she is dealing with it - you need someone you can be confident in on your case).
Most likely, your sellers are just badly organised/stressed/dumb. And haven't got their act together. The only time I experienced this was with a widower who just couldn't cope with the enormity of moving - when someone talked to him and organised the removal van he moved out like a lamb. Perhaps your sellers are just dysfunctional and/or lazy and either need someone (not you) to read them the riot act or organise their move for them. (And make sure they foot the bill).
Worst case scenario, and very unlikely, you're looking at having to evict them. However, neither you, your solicitor, or mortgage company will want this. And nor will your sellers if they know they can't mess you around. If they don't move out, they should be liable for your losses - your solicitor needs to lay this on a bit, then they will release that working hard to find removal men or hiring a van and putting their stuff into storage really will be their cheapest and easiest option!
Do pursue your solicitor and the estate agent (but be wary of agreeing anything with the latter, just take soundings and remember they are acting for the seller). It may just be down to miscommunication, a half-useless solicitor and a blip. Chances are it can be sorted out by the sellers getting removal men set up, going into storage, whatever, and if absolutely necessary staying in for a day longer under a licence arrangement, but it needs to be agreed and documented so that you and your mortgagee are protected. Don't be fobbed off. And try not to take too much of the heat yourself.
Stressful at the best of times. Let the people you pay to sort it, sort it (providing they are not completely useless/negligent, in which case, you may have to think about instructing new ones). That should also mean that you can maintain some sort of relationship with the seller....
Hopefully, things will sort themselves out. Fridays are the worst times to complete. Come Monday, things may look very different. Good luck, and keep the gin handy.