I realize this thread is very old, and probably the original poster is long gone, but nevertheless I wanted to post how I managed to get this game running fully, with videos and everything.
First off, I'm pretty sure that modern DirectX screws up the video playback within the game, this would explain why even on my previous 98 SE drive the game would run but not play videos. Since for any Doctor Who fan the brand new (for the time) material recorded for the game is the highlight of the proceedings, this clearly would not do.
Enter Virtual PC 2007. This is available freely from Microsoft's website. After installation, I began the new Virtual Machine setup wizard and created a VM named Windows 95, and gave it the attributes of 128 MB RAM. The system automatically defaults to emulation of an S3 video card and a Soundblaster 16 sound card. I then started the VM for the first time.
Then, I placed my Windows 95 CD in one of my drives, and selected the CD menu at the top of the running virtual machine window and selected the CD-ROM containing the Win 95 CD. Then, I selected the Action menu, clicked reset, it autodetected the CD, and I then proceeded to install Windows 95. Once that was done, I inserted my Destiny of the Doctors CD into the drive, booted up the VM, and went into the VM's My Computer, opened the CD, and installed.
The only other modification I had to make was to keep my desktop res within the VM at 640x480 and then, after starting Destiny of the Doctors, going into its Settings menu and changing the game's res to 640x480 (if you don't do this the playing field in the game will be the wrong size and cut off part of the screen). I also set the VM to always start in full-screen mode.
And it works like a dream.