I have been trying to get Fallout 3 to work on GOG but have had no luck. I did the two most common solutions, which is to make sure the presentinterval is set to 1 and the fpsclamp is 60. I did this in both the GOG file and the falloutini in the documents. I did initially have the game on steam, so are the files interfering or something? Everyone says that the GOG version doesn't require mods so I haven't tried installing any.

Install mods.

FNV GOG has the same problem and i fixed it with an anti-stutter mod.

Like @Vozhd said, Fallout 3 and New Vegas will always stutter no matter if you play on a console, through Steam, GoG or whatever. The only solution is to use mods. Taken from one of my other posts:

To fix the stutter you need at least the following mods:
- Stutter remover. This fixes 90% of the stuttering. Cap you FPS at 60, sutter and bugs seems to be inevitable above it. Set "bReplaceHeap=1" and "iHeapSize = 450" in the ini file
- Unofficial patch: fixes so many crashes, bugs and a few stutter as well
- NVAC: fixes general crashes, although it doesn't reduce stutter I highly recommend it.

Good source of info here. Not the OP, but thanks!