Delays are part and parcel of buying and selling I'm afraid. There are some ways to speed the process up. You need to be nice, ask lots of questions and write down the name/date/time of everyone you speak to.
You should call everyone you're allowed to:
Your solicitors
The estate agent
Your mortgage company
The housing association
You can't speak to the vendors or their solicitors. You can ask for another visit to measure for curtains, and speak very nicely to the vendor then.
Write a list of everything that needs to happen and where each thing is up to.
Eg: If your solicitor says you still need to receive and sign off on the PIF, you can ask when it was sent to the vendor to complete. The estate agent does that bit, so call them. When did they receive it back? When was it sent on to the vendors solicitor? When did they send it to yours? When did they send it out to you?
Somewhere within that chain, the PIF is stuck. Find the PIF and move it along. If you are told the vendor posted it back to Estate Agent yesterday, call the EA. And keep calling. If 3 days pass, ask the EA why they haven't chased the vendor already.
Do this with every part of the process, come back to the lovely MNers on the property boards and they will guide you on what questions to ask next. Micro manage in an annoying but really friendly way, overly friendly, as being micro managed will piss them off. But you can shave off a good 3-4 weeks by doing this.
Have all your 'facts' written down. If Julie told you the searches were submitted on 15/1 and you speak to Bob who says you are still waiting for searches to come back, ask Bob when they were submitted. Triple check everything is being done when people are saying it is being done. Conveyancing is so time consuming and lengthy because of how many parties are involved. It's easy for someone who has forgotten to do something to blame it on another part of the chain. Detailed notes give you the opportunity to call them out on it if spotted and also let the people you're dealing with know you cannot be bullshitted.
Problems to anticipate:
You said the vendor is in a nursing home and the daughter moved out. Clearing possessions is one issue, but not one that will delay things too much. The vendor may no longer have the capacity to make financial decisions or sign legal documents. This may be why the PIF is delayed. If the daughter didn't already have power of attorney, it is then too late get it. They will need to get a Court if Protection Deputyship to sign any of the documents, you can ask your solicitor if this is an issue.
Money can also be an issue in this situation. The above needs to be paid for before the sale of the property, along with some documents from the Housing Association in terms of lease and any covenants. There are some other documents to do with proof of no service charge arrears that can sometimes be requested. If the vendor doesn't have this money yet, or the daughter simply cannot access the funds to pay for it, the whole thing grinds to a halt. Depending on their own financial circumstances, you could be waiting for them to save up the money to pay for these things. If this is the case, you may be asked if you will pay the fee on their behalf and the purchase price would be adjusted accordingly. That is bad news and your solicitor has to ask you but should advise you not to. They are fees you can't recoup if the sale breaks down.
Once you know more about the vendors financial position, you should look in to the other flat in the building. Rather than ask for an 8 week deadline, ask in depth questions about the vendors position and motivation for selling. Not why they're selling, but how keen are they to get the deal done.
Best of luck, keep cool and calm. You will get to the end of this. Don't let the process ruin how you feel about your first home.