Welcome to Android Central. Thanks for registering.
I have a Samsung Note 3 with the now older Touchwiz, and I am not familiar with Lollipop yet. Others will help you better here.
So you have had no Mobile data since upgrading, only wifi. Nothing you do gets you connected. You say 3G but that may have a 3G carrier / network.
Bear in mind a full OS update like I discovered with KitKat resets or turns on a lot of Google settings like syncing that you may have opted out of previously, and it takes a few days for syncing and battery and data use to settle down.
You can / may have checked you have apn settings stil intact, they should be, in Mobile Networks - Access Point Names.
You could look at About Device - Status and see you still have the carrier showing, but no reason not to be.
If you don't get more advice here (some people think I know what I'm doing
)
I would recommend trying two other areas on AC with this question, where experienced site ambassadors are more likely to pick it up quickly.
Android Central Community - Introductions (& General Help and How To); and also the Samsung S5 forum.
It's probably something simple that I can't think of right now, but put more information out if it's relevant.