Have you tried Compability mode?

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with different compatability mode settings

Yes

I'm having the same issue. (subscribing in case someone finds a solution)

Edit: to save time, here is a list of tests that have been run with no success. Please check if your solution is listed below:
changing graphics mode to basic (or any other setting)
Changing to windowed mode (both through res_config.exe and through manual mode)
testing different resolutions
testing x2 scaling
Installing Fallout Fixt and running it instead
enabling or disabling Fallout 1.1 Hi Res patch
Verifying DirectPlay is active
Changing compatability mode for falloutlauncher.exe, falloutW.exe, and falloutwHR.exe
giving administrator privilages to executables listed above
uninstalling and reinstalling the game
Restarting computer and steam
Performing a "Verification of Integrity of Game Files" through steam
Running the game in both Normal and Classic mode

In your Fallout folder run f1_res_Config.exe. Select "Graphics mode: Basic" and your resolution and then "Enable" at "Fallout US v1.1". That worked for me. You can also try to enable DirectPlay under Control Panel > Programs > Enable or Disable Windows functions > Legacy something > tick DirectPlay. That works for me without problems on Win 10. Same goes for Fallout 2.

*subscribing*

Hey, I was having the same problem and this fixed it. Goto preferences for fallout 1, select local files tab and browse local files. then right click on FalloutLauncher.exe and select Properties. Under Compatability tab check compatibility mode and select windows 7.
Hope that fixes it for you guys.

Unfortunately, like OP said, different variations of compatability modes don't help the issue. I've tried variations of compatability modes for the launcher, the "FALLOUTW.EXE", and the "falloutwHR.exe" files, with no change in results. Also, like OP said, activating the Hi Res Patch 1.1, or playing with the graphics settings doesn't help. I've verified that DirectPlay was already active.

I've also run a "verification of game file integrity", and, though there were two files that needed changes, it didn't help with the problem. (Needless to say, it's been several days since purchasing the game, so I've powered down both steam and the computer several times.) I've also uninstalled and reinstalled the game, with no change.

Edit 1: tried running as administrator, problem persists

Sorry, I understand as I fiddled with it for a long time. But that's what finally fixed mine. Hope you figure it out

Start the game in windowed mode

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with various different resolution settings (my native res windowed, diff res windowed, my native res streched X2, etc.)

Jokerman, I appreciate that you're trying to help, but it saves time if you make sure your suggestion hasn't been tried yet.

I've been looking everwhere and I still can't figure this out. I've tried all of the above to no avail. Any new ideas?

I gave up and got a refund, myself.

I think Windows 10 is simply uncompatable.

Happened to me. Had another application running at the same time. In higher resolution. Exited the other application and Fallout worked again.
f1_res_Config states: Basic mode. Windowed, 1600*1200, 32bit, 59Hz, Hi-Res Disabled.
But game runs in full screen.