You’ll most likely find someone at the jet bridge/bus arrival area for your inbound with your flight name/number on a sign, who may/may not escort you straight to the new gate depending on how tight the connection is, and how far away the gate is. Leaving the lounge an hour before is still very comfortable timing so as not to rush, you can do it in much less time. If in Business it will likely be a small group, or personal escort. I’ve also had this a platinum in economy.

As other posters have said, as long as they know you are there they’re unlikely to offload you as it is more complicated for them.

What I would say on that tight a connection though, is I hope there is no delay to your inbound, as they can be a bit trigger happy to offload you from your next flight and rebook if they feel you will be late. If all goes to schedule you’ll be fine. I;ve had shorter connections frequently with no baggage or boarding problems. Either way, you will get to where you want to go.