Most solicitors use a search agent, so that's not unusual. If there's a delay, then it will be either with the search agent or the Council, so not much you, your solicitor, your buyer's solicitor can do about it.
It is highly likely that all the searches are back, except for the Local Authority Search. Unfortunately, should any of the searches reveal anything concerning, then the buyer's solicitor will need to make further enquiries in relation to it. So, that also can delay the process. There is a search indemnity policy they could buy, but while that would protect them financially, it won't show them what the searches reveal.
If you are relying on your buyers to part fund your purchase, then you and anyone else in the chain will have to wait.
You are only really ready for exchange, when all parties are ready.
Hopefully there will be no problems/issues revealed in the searches and you will proceed quickly and in time for school.