Coal searches are normally back very quickly (I'd say hours) as they are computer generated and often returned by e.mail these days. Depends how your conveyancer did them eg online (far quicker) or through search agents or the more old fashioned paper and postal way.
Ask your conveyancer for a full update as to what, if anything, remains outstanding and the plan for rectifying it and when the soonest is you can exchange and complete (make sure they doublecheck with the seller's solicitors first).
Agents can be very good at chasing both sides up and communicating issues and helping resolve them so they are often worth a try.
Conveyancing can be done incredibly quickly these days.
Picking a date and getting it agreed with the sellers can be a good plan before putting it to your separate conveyancers to doublecheck it is feasible and anything necessary is done and ready (eg getting a mortgage redemption figure and drawing down a mortgage advance).
From the sound of it they are waiting for all search results (and any outstanding enquiries) to come in before they send their report on title and documentation for signing.
If you use e.mail, that can speed things up as I'll tend to scan and e.mail whatever I can to get things out quicker.