I would believe from your description that you are not creating a local storage key on your mobile game.
You did at some point on your test computer, so the key exists, and you apparently have code that DOES save the value into the key.
On the mobile though, you possibly only try to load a value from a key that does not exist.
As I explained in the tutorial, you need, the first time you execute your game on the device to create the key and populate it with a default value.
So in a "On start of layout" in the first layout of your game, have an action that checks if your local storage key exists (Check item YOURKEY exists).
If it does exist, (event "On item YOURKEY exists") then you set the value of your global variable score to the value of the key YOURKEY in the local storage.
If it does not exist (event "On item YOURKEY missing") then you start by creating the key action "Set item" YOURKEY to a value of 0 and set the value of your global score variable to 0 as well.
If the rest of your code is correct (and that you are always using the correct key YOURKEY name), all should work as intended.
Otherwise, do post your capx, that will be the easier way for other people to possibly see where you are going wrong.
If you feel it is too complicated, take your time.
Read several times, over several days the various explanations (and tutorial) and it should become clearer after a while.
First time ever ? (no key exists) => Create a key
Not the first time ? (a key exists) => Set the value of the variable to the value stored in the key
Ending the game ? => Save score in the key before the user has exited.
At worst, delay the exit so the score IS saved.