In fairness, you've waited until quite late in the day to start doing your searches and enquiries. I know everyone does it differently but on every house I've bought/sold, the searches have been requested much closer to the start.
Your survey is done, and your mortgage offer is in, and there's no chain. So that's all positive.
Searches usually take about two weeks - but in some areas it's taking much longer, could be up to 40 working days.
But let's say your searches come back in two weeks. Your solicitor isn't going to pounce on them the minute they arrive because your sale has to take its turn in her caseload, just like all the other people buying and selling property. Your solicitor will then need to go through the searches and identify any potential areas of concern. It's normal for enquiries to be raised off the back of the searches. So they will need to write to the seller's solicitor with their enquiries. The seller's solicitor may need to contact the seller to get answers. Seller replies to their solicitor. Their solicitor then writes back to your solicitor. Your solicitor reviews the answers. If all is OK, contracts can be sent out for signature. Once they're returned you can agree exchange and completion.
Can you see why 4 weeks is ambitious? There's multiple touch points, and every time a piece of correspondence is received, there will inevitably be a small delay as your solicitor gets round to it - hopefully no more than a day or two. Also depends on how quickly the seller answers any enquiries.
We're in the middle of selling our house now - process started on 4th August. We are - hopefully! - at the point of just finished enquiries and the contracts are being sent out so we're nearly there. But we have no chain, an incredibly motivated buyer, and we both have shit-hot solicitors who don't hang around. Even with all of that it's going to be around 8-9 weeks minimum, start to finish.