My players track things like HP and Spell Slots on scrap paper. They have a character sheet which they sometimes refer to... and a scrap sheet with frequently changing values like HP or Spell Slots or Death Saves. If they have Lucky, they also have the word Lucky written on that sheet, and will add some tally marks to it as it gets used. That's pretty hard to forget. It's one of 3 or 4 things staring at you from that blank sheet of paper, along with HP and Death Saves.
During the game, the player rolls the dice and observes the roll. If they are unhappy with the result, it's on them to say: "I use one of my three uses of Lucky". They then do the re-roll, and tell me the new result. They mark off one use of Lucky.
If they're influencing one of my (DM) dice rolls, they will let me know after they see the roll. I throw all dice in front of the players (always; regardless of Lucky or not; there are no hidden rolls at my table, unless I'm actively trying to make them paranoid). It's on the player to announce immediately whether I need to re-roll, before any other effects are applied (damage rolls, reactions, etc).
If the player forgets they have Lucky, there are no take-backs. Once damage is applied, or we've moved on to another character's turn, their window is closed. That's not intended to be harsh; it's intended to keep the game moving. If the player forgets, then maybe they weren't as Lucky as usual for that particular roll. I promise you that they'll remember better next time.
I've never had a problem with this at the table, for any type of re-roll or double-roll. Could be an Advantage roll, a burned Inspiration point. It doesn't matter. You have your window; you use it, or you don't. All of my players know the expectation and respect it. It makes for quick play and equitable adjudication. It also lowers the workload on the DM.
I'll make an exception for a complete newbie, of course. But that's not about the Lucky feat specifically. You should be taking care with nearly everything (opportunity attacks, saves, skill checks) regularly for a new player, and making sure they understand their options.