Solicitor here, I work primarily in conveyancing so I hope I can help.
The usual buying process is as follows:-
Initial paperwork and payment on account is received by the client (you)
We carry out our due-diligence (ID checks, source of funds checks etc.) and then request contract documentation from the seller's solicitor.
Once received we order searches. Previous posters are correct in saying local authority searches take varying times to be received. In my area, we get the environmental search result almost immediately, the water a drainage search around a week later and the local authority search in around 3-4 WORKING weeks.
The next step is where solicitors vary. I personally raise initial enquiries on my clients' behalf before I receive all the search results. So when I receive contract documentation, I review the same and raise enquiries on the Title (deeds) and the protocol documentation completed by the seller (the property information form and the fittings and contents form). I also send the protocol documentation to my client along with a plan to the property for their review and to give them the opportunity to raise any personal enquiries they may have. While I'm waiting for the seller's solicitor to respond, I'm also waiting for the search results to come in. Some solicitors wait for all the search results before raising enquiries. It really is personal preference. Of course, once the searches are in, if required I'll raise additional enquiries on these.
Once all enquiries have been responded to and I'm happy with the same, I'll prepare the report on title and send this to my client with documentation to sign. The report basically details everything I've learned about the property through my enquiries and the search results and puts it all in layman's terms. If you're happy with this, we'll confirm this to the seller's solicitor and begin to discuss completion dates.
I will caveat this by saying not all solicitors and definitely not all firms are the same. We're currently experiencing a huge problem in the industry with what we dub as "factory firms" who take on far too many cases to be able to effectively manage their load and also client expectations.
If you're unhappy with the service you are receiving, make a noise about it. There may be a perfectly reasonable explanation, but you should be kept in the loop as to process of a matter which constitutes the biggest purchase you'll likely make in your life. If there is a delay, they should be updating you.