I have tried the usual things like compatibility options, but it doesn't help.

Any ideas?

Thanks.

Maybe this is connected to Windows' audio architecture. Since Vista, Windows doesn't support sound hardware acceleration for DirectX anymore. All 3D sounds are processed by your processor now. I suspect RCT 1 + 2 to use these old DirectSound3D features, which could now stress your CPU somehow. This could result in stuttering.

You could try running the game through a DirectDraw wrapper, which in certain cases uses less resources, thus there's more capacity for sound processing (here's one to run the game in windowed mode). Also reduce your PC's workload to a minimum (turn of anti-virus and other programs while playing)

You can also try to installing a solution of your sound chip manufacturer. Some of them are providing software to bring back the old hardware acceleration features under newer versions of Windows. This comes in handy for some older games.
- Creative Labs' solution is called ALchemy
- Realtek's solution is called 3D SoundBack
- C-Media's solution is called Xear3D EX
Which one to use depends on your sound chip.

Thanks for the reply.

Strangely I ran both slowed down with Mo'Slow (an 98/XP slowdown program for older applications) and the sound and video are now perfect.

Nice find and it makes perfectly sense. Slowing down the game's calculation cycles frees capacities of your processor. Parts of the game seem to stress it, so that there's not enough processing power left to play the sounds properly. It could also be that the bottleneck results from constant reloading of data, which when slowed down is solved the same way.