As I say, I've only seen one Bond film, but Alison Doody's first film was A View to Kill. A View to Kill's cast also included David Yip, known to Indy fans as Wu Han.
Also, Vic Armstrong, Harrison Ford's double for the initial trilogy, doubled for George Lazenby in On Her Majesty's Secret Service, which makes him, apparently, the only man to portray both Indiana Jones and James Bond. The other prominent stunt man from the trilogy, Pat Roach, was in Never Say Never Again.
And then there's Christopher Lee, who plays Scaramanga in The Man With the Golden Gun and Count Czerin in the Austria 1917 segment of Adventures in the Secret Service. Max von Sydow was Blofield in Never Say Never Again and was Sigmund Freud in the earlier Vienna 1908 segment of Perils of Cupid. Julian Fellowes was Churchill in Love's Sweet Song and appeared in Tomorrow Never Dies. Michael Kitchen, who portrayed David Lloyd George in Paris 1919/Winds of Change was Bill Tanner in GoldenEye and The World is Not Enough.
Philip Stone, our Captain Blumburtt, had a bit part in Thunderball. Ravil Isyanov, Sergei of Petrograd 1917, also from Adventures in the Secret Service, had a bit part in GoldenEye. Terry Richards, the Arab Swordsman from Raiders, apart from portraying the Bavarian in Austria 1917/Adventures in the Secret Service, had a bit part in Tomorrow Never Dies.
Doing stunts in GoldenEye were Perry Davey, who was the clown Zoltan in Phantom Train of Doom, and Graeme Crowther, who was Harper in the same Young Indy film. Crowther also did stunts for A View to Kill, The Living Daylights, For Your Eyes Only, The World is Not Enough - and both Temple of Doom and Last Crusade. In a similar vein is Sean McCabe, who seems to have been a stunt double in almost every Young Indy film, had bit parts in two of them, and did stunts for GoldenEye and The World is Not Enough. Nigel Fan was Ah Pin, the cart driver, in Peking 1910/Journey of Radiance, and did stunts for Tomorrow Never Dies.
Wow, I just spent over an hour doing that. I'd put money on there being at least a handful more, though, especially for stuntmen. The ones whose names I happened to check were more flukes than anything else, and there's probably one or two notable character types in the Young Indy's that I missed too.
Oh, and if anyone were to start a Sherlock Holmes thread - geez, you'd find a lot of crossover. Also Shakespeare adaptations, obviously.