I know it is an unpopular view on MN but I have done well by asking for recommendations from the EA. (A proper EA, NOT Purple Bricks).
EAs are the people in the mix who alongside you want to get the sale over the line as fast as possible because they are the only ones paid on results / commission.
They have been involved in the progression of numerous sales and can see which solicitors have been fast, efficient and pro active in solving any issues.
Hold ups occur if there is a shortage of surveyors, your mortgage co can be slow, searches taking ages and vendors not replying to enquiries, returning their paperwork.
EAs are good at chasing all parties (if they are good at progressing the sale). Ask them for a weekly update.
Have your vendors found somewhere else to move? The biggest hold ups atm are vendors not being able to find a new house. And if they do, being part of a chain. And you can’t control that.
Are they buying a probate property? If so has probate been granted yet? It can take ages. Are they buying a new build? Developers are asking people to commit to ‘long stop’ completion dates because they can’t guarantee that the house will be finished.
All this could cause much bigger delays than solicitors.