Hello
I think for & you can use Garamond:
http://www.whatfontis.com/Original-Garamond-BoldMultilingual.font?text=-
and for JKP you can use Futura Medium:
http://www.whatfontis.com/Futura-Medium-paratype.font?text=JKP
Alex
I found it using: http://www.whatfontis.com - Identify the font you are looking for!

1) As previously said, very generic, nothing really specific. Suggestions by Alex look fine to me.
2) sorry, don't understand what you want exactly... Do you mean "Bond Street"? Because I wasn't able to spot any "Brook Street" lettering anywhere... For the rest, Smythson identity has been designed by Saturday. They may use some bespoke typeface for all their materials. On their website, thery're using Brandon Grotesque and Electra Display but I suppose it's not the one you're after.
For the rest, can't get out of my head that main woodmark SMYTHSON could be easily done starting with some Futura Extra Light manually extending differently each letter. It's some sort Engravers Gothic/Blair/Sackers Gothic extended.
Some more "Futura-like" (letters needing to be stretched: Semplicita, Agenda, Contax Pro, Okojo
Some "Extended Engravers Gothic": Aviano Sans, Montag, Sweet Sans, Luxury Gold, Capline, Zeppelin (41), Doublewide, Adrianna, Biondi Sans, Halogen, Idlewid, Burin Sans, Ronsard Crystal

Anyone? (Please see the link I posted above with screenshots of "Brook Street" - Smythson)

Thanks HVB to confirm what I've said. So definitely custom typefaces exclusively for SMYTHSON. So, you could still have a look to my previous suggestions.

Here's a closer look to Brook Street embedded as webfont on Smythson website. Definitely very close to some Engravers Gothic/Blair/Sackers Gothic, for the caps at least. Lowercases is serifed though, the very big difference.