I've played clarinet before bassoon, and your fingers that are not the thumb are somewhat similar for going up and down the scale, but I don't think there is another instrument that gives the thumbs such a workout as the bassoon does. Also once you get out of a register into another one it all tends to get quite different.
The contrabassoon is closer to the bassoon than anything else you will find, but it still has some irregularities. I think the tenoroon shares the fingerings exactly, but it is practically the same instrument.
Bassoon fingering from low F up to the D an octave and a sixth above is very similar to fingerings used on flute, recorder and oboe (I don't know about clarinet or sax as I've never attempted to play either). Of course there are some differences between these instruments, but if you've learned one then the fingerings will feel familiar on the others.
However, if you use broadly the same fingerings on a flute, oboe or soprano (descant) recorder what you actually get is a scale of C. On an alto (treble) recorder you get a scale of F, as with the bassoon.
Below this range the bassoon uses thumb keys to get you down to the bottom end; and above it the fingerings for the different instruments start to diverge.
for some reason some bassoon fingerings remind me of the flute fingerings....
like G on basson is kinda like the flutes D.... and.. bassoon F (low one) is kinda like flute F. When i switched from flute to bassoon i found the fingering to be the easiest part.. erm.. except for the low low notes xD.
@Heckelphon: I think he was asking outside of the family and even within the family its a haywire system. Tenoroon and Bassoon start similar but end entirely different in the high registers and such because the bassoon has millions of complicated permutations for fingerings. Contrabassoon you make up your own fingers in the extreme register after you pass the e right after middle c and quint and octave bassoons are the same story as tenoroon.
In short this beast can't be compared to most other woodwinds.