All credit for the methods used here goes to @lordtobi, who posted a few different tricks around the Alpha Team subforum. All I've done is collate his work into one simple tutorial with screenshots, so please direct any appreciation his way!
1. Insert the LEGO Alpha Team disc into your PC. Open This PC/My Computer, right-click on the disc and select “Open”.
2. Select all files on the disc, except from the "DirectX" folder, and copy them.
3. Create a new folder somewhere that you have permissions (e.g. Desktop, Documents, but not Program Files). Paste all the files inside this folder. Note that in the previous screenshot I have a “desktop.ini” file selected, this doesn’t need to be copied.
4. Inside this folder, create a new folder called "DirectX". Download the .dll files here, extract them, and place them inside this DirectX folder.
5. Back in the main folder, right-click on “Setup.exe” and choose “Properties”. In the window that pops up, select the “Compatibility” tab and set it up as shown below. Make sure you press “Apply” before pressing “OK” or closing the window.
6. Run the “Setup.exe” file in this folder. Follow all the steps, clicking OK when informed that “DirectX 7 will now be installed on your computer”. The application may freeze temporarily, just give it a minute. Click “Finish” to close the setup program when it is complete.
7. Navigate to the folder where LEGO Alpha Team installed. For me, this was: C:\Program Files (x86)\LEGO Media\Games\LEGO Alpha Team
8. As with the “Setup.exe” program before, right click on the “Launcher.exe” file, select “Properties”, then go to the “Compatibility” tab and set it up as shown below. Do the same for the “LEGOATeam.exe” file.
9. To run the game without requiring a disc, first run "Launcher.exe", then press "Exit" when it starts.
10. Now copy the “Config.txt” to somewhere you have permissions (e.g. Desktop), and open it in Notepad. In the below screenshots I’m in the install directory, but you probably won’t be able to save from here.
Remove the “DiscPath #:\” line, then under the DisplayGUID line (I’ve blurred the code as I wasn’t sure if it was important), add in “VerifyDiscVol false”, including both spaces. Save the file and close it, then copy it and paste it back over the “Config.txt” file in the install directory, replacing it.
11. To keep the intro video, navigate back to the folder created in Step 3 (where you pasted the disc’s files), copy “lmiintro.avi”, and paste it in the install directory. You can now delete the entire folder created in Step 3.
12. Finally, to set your custom resolution, run the game and adjust the settings to populate the configuration file. Then select the “Game.txt” file in the install directory, copy it somewhere that you have permissions, and open it in Notepad. Again, I show the file in the install directory here, but you’ll want to have it somewhere like your Desktop.
Change “BPP ##” to “BPP 32”, then change “Width ####” to your desired width and “Height ###” to your desired height. Remember to leave two spaces between the text and the numbers. Below is my modified file.
Save and close it, then copy it and paste it back over the original “Game.txt” in the install directory, replacing it.
Congratulations, you’ve installed Alpha Team and are ready to play it on Windows 10!