I have a '67 Bird with a 3 speed manual (saginaw) trans. I don't suppose I could be lucky enough to swap in a 4 speed gearbox and use the same driveshaft/ bellhousing & clutch/ crossmember, etc. I know the shifter is different..

I highly doubt it, but thought I'd ask just in case there is some interchangability...

I switched from the Ford 3 Speed to a Muncie.
I kept the bellhousing, crossmember, and Driveshaft, but I had to change the front yoke, which was no big deal. I don't remember if the clutch was different, because I bought a new one for the swap- you should at least change your disk anyway while you have it apart.
Overall its very easy. However, if you have a console, not that it may be hard to get a modern shifter into the console- you may have to use an original, which are pricey.

good to know, thank you!