This is a common problem. If you want to run Lemmings 2 on any version of Windows past Win98, it is strongly suggest that you use a patched version that fixes the problem you described, and also to try running it in DOSBox instead of directly in Windows.
The patched version is available
here. Run "L2-fix" instead of "L2" to run the patched version of the game which fixes the "can't get to next level" problem you described. When you run L2-fix, press ENTER to get past the text it shows when it starts so it goes to the actual game.
While it may not be absolutely necessary, if you still have this or other problems with the game when running it in Windows, try running it inside the "DOSBox" program, a free emulator of DOS (and also emulation of old computer hardware from the DOS era). You can get the latest version
here. You will probably want to read their instructions/readme file before you begin using it, as there are some additional steps you need to do before running a game (eg. mounting a directory).
