I am in London two weekends in September, so I will keep an eye out for this meet up.
I'm sure there are people here more familiar with Jermyn Street than I am, but I do visit there reasonably frequently and my observation would be that some of the shops are quite small. I'm thinking of Tricker's here, for example. There wouldn't be room for a decent sized group to stay there for long, if at all. The Crockett & Jones shop is bigger. In both shops I think prior contact with the staff would be essential. The Tricker's guys are old school and somewhat curmudgeonly (in a very polite and knowledgeable way, LOL). They usually have some interesting bits and pieces in the Jermyn Street shop and will usually be happy to show that stuff off to people who are interested. Last time I was in Crockett & Jones they were possibly more friendly, but I feel like it's because they're a more sales-orientated shop, they'll pitch to you from the moment you go in to the moment you leave. I find C&J the same in Burlington Arcade, maybe it's the way they're training their staff. I just prefer a more hands off approach.
In terms of pubs there are two I go to near Jermyn Street, but I should clarify that I am a craft beer fan and I like places that have a decent variety.
The Lyric is actually technically in Soho but it's walking distance (5-8 minutes) to Jermyn Street and is a decent compromise between old school London boozer and craft beer options. It has the Victorian interior but a modern line-up of drinks offerings.
The other place I go nearby, similar walking distance, is the London Beer House, which is possibly an even better craft beer bar but is less well appointed. It's on the corner at the top of the Royal Opera Arcade and has seats outside. Most of the seating is beer barrels or sort of lawn furniture, quite light tables and chairs. If you don't like craft beer there is probably no reason to go here.
I took a look at some of the other pubs mentioned and I think The Red Lion looks like a decent option although I've never been. It's a Victorian horsehoe layout and looks like there is plenty of room. All London boozers get pretty packed between 5pm and 7pm with the after work crowd, in my experience.