I want to transfer the save game back and forth between my laptop and Desktop but cannot for the life of me find where the savedata is.

On Windows 10 it looks to be C:\Users\<username>\AppData\LocalLow\Playtonic Ltd\Yooka-Laylee

C:\Users\(username)\AppData\Local\GOG.com\Galaxy\Applications\49987265717041704\Storage\Shared\Files

or search for ddda.sav on AppData

If you are running the game on Windows without GOG Galaxy you can use this path:
%APPDATA%\..\LocalLow\Playtonic Ltd\Yooka-Laylee

This is essentially the same path as CountDuckula supplied, but using the path above can be used without having to replace the username, which is a little easier.

If you are running with GOG Galaxy, just enable cloud saves to sync between the computers.

- message deleted -

On Windows 7 64-bit NOT installed through Galaxy client but with the downloaded offline setup file the saves were in this directory on my system.

C:\Users\(Username)\Documents\Activision\Gun\saves

The save files were named exactly as I named them in the game; for example if you saved in the game and named one of the saves Battle there would be a file named Battle with no extension. This makes using the technique of searching for ddda.sav irrelevant for users who have the same installation I do.

I hope this helps someone; I had to dig to find this out and thought it deserved mention along with the other possible solutions.