I just downloaded both Fallout and Fallout 2 however when I start the game I can hear the music but the screen stays black. Also, the launcher window graphic is distorted. Anyone else experienced this problem or have a fix? Grateful for any help. I am running Windows 10 on a Dell Inspiron 15 laptop, I have no issues running any other games from GOG.

For Fallout 2 try Killap's Unofficial Patch or Restoration Project (RP):
http://www.killap.net/

HighRes and Sfall mods/patches are included so it can help...