How to slow down an application?

Not very easy task in modern systems. In past you could use software that wasted resources...

Try CPUkiller or NTsleep.

You can try enabling vsync in your video card, or using a software to limit your FPS, like FPS Limiter.

omg laaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaagggggggggggggggggggggggggggggggggg

AFAIK Lemmings was a DOS game, so use DosBox and look at the documentation to adjust CPU cycles.

I have no idea if this would actually work but:
Try running the game in Windows 95 compatibility?
Try giving it less priority through the Task Manager?

Really, I don't know if this would change anything, but always nice to try. ;)

Dosbox or force cpu to min P-state:
On Windows 7 go to Control Panel > Power Option > Balanced > Change plan settings > Advanced settings > Processor power management > set 5% to min and max.
Of course you have to set 5% (min) and 100% (max) to restore the default settings.

Either use "Lemmix" instead or "Dosbox" + "Virtual Sound" the original game.

You should limit FPS. I don't know how

There are programs that can artificially slow your CPU down by giving it busywork. This will slow down everything, not just your desired program, and doesn't work particularly well on newer systems which can automatically give CPU priority to the program currently in focus.

I remember that I read about a way to alter the old exe files for Command & Conquer Red Alert 2, which allowed it to run at a proper speed on Windows XP and up. Consider looking into that. I also remember someone made a patcher exe for the original version of Majesty, which did the same thing.

In the end, the very best way to get it working properly, if you can't run it on a real 486, is to run a virtual machine. I recommend VirtualBox. There is a preconfigured VirtualBox and DOSBox here which, while intended to run the Stars! 4X game on a modern PC, will also run anything else on an emulated WIndows 95/98 box. I still use it to play Descent 1+2, Magic Carpet 1+2, Arcanum, and Castle of the Winds.

Have you tried opening task manager and right clicking the .exe process, choosing Set Affinity, and making sure only 1 CPU core is ticked?

:D Lemmings. That game is great.

Also, this as a stop gap until you can get your version running properly ;). Sorry I can't help otherwise!

Haven't seen CheatEngine mentioned yet
I don't know if it would actually work, but I used it once in the past to slow down a game (the "speedhack" is a specific function of the tool). Probably not be a good idea to run it while running any Valve/VAC-bannable games though (no idea if they really detect it or not; just doesn't sound to me like a wise thing to do.)

Cheat Engine can slow down most apps.

I had a similar problem (it actually was the opposite as I had to SPEED up a game which was going slowmo for no apparent reason) and I fixed it with Speed Gear