I bought Tab Pro subscription on my desktop and can't get access on my Android/iPhone/iPad

You need to be logged in under the same Ultimate Guitar account which was used to purchase Tab Pro subscription.
Tab Pro subscription is always attached to a certain Ultimate Guitar account so it all makes sense, doesn't it? 

Just make sure that you've downloaded the correct app. 
Tab Pro for Android: https://play.google.com/store/apps/details?id=com.ultimateguitar.tabprofree

Tab Pro for iOS: https://itunes.apple.com/us/app/tab-pro-worlds-1-guitar-tab/id535076358?mt=

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