Legal industry insider here.
A) Don't choose on price. Low-end, high-volume conveyancing firms pile loads of work on stressed out/burned out staff who often aren't exactly loving their jobs. You might get very lucky but atm with the stamp duty deadline approaching conveyancing firms are even more busy than usual. Avoid 'fixed fee.' This generally means you guarantee how much they are paid but they do not guarantee how much work they do for you, or how quickly.
B) If you have a complex transaction or fast deadline, go for a firm that is 'high end' enough to be motivated to keep their stellar reputation and their excellent Trustpilot reviews. Expect to pay above 4k for sale and purchase. Anyone recommended in Chambers and Partners or Spears is probably better than a firm that isn't but has a lot of Trustpilot reviews. The legal directories are very well researched.
C) There are national chains but I also look at whoever is the 'high end' firm in your local area. They have more to lose by doing a bad job for people in their community than a branch office of a big national firm does. A local firm that does not have say a central London HQ could well be just as good as a national firm and also cheaper. I would prefer a local firm whose website shows me the partners also have some national firm experience. That suggests they were good enough to get hired by a big firm originally and moved to local for lifestyle reasons.
D) Go with who your agent recommends if your agent is honest, efficient and seems highly motivated to manage your chain.
E) If your agent is recommending someone, get a competing quote from a comparable solicitors' firm to make sure not too much commission is changing hands, if any. I just used my agents' recommended solicitor because I knew the firm but I also argued them down on price after getting a competing quote from a firm that I know is in exactly the same bracket. Not sure how you would know which firms are in which bracket automatically (I know cos of work) but one tip is to use Linkedin to research where partners at a firm you are considering have worked previously.
F) Finally, its always great to ask friends in your local area. Be sure to try and use the same person they used and liked because different solicitors work to different standards in all firms.