While the OP had only a carry-on and thus did not have to deal with baggage claim, I would like to share our recent experience for those who do.
Due to unforeseen circumstances, we had to alter our travel plans. We had planned to be in London for a few weeks, then fly to Amsterdam and then take the train to Rotterdam. That didn't work out, but we had two sets of tickets that could, for the fees, be saved. But our timing was tight.
Our Norwegian flight left Orlando Tuesday afternoon, an hour late. The pilot made up half the time. We were due to arrive at Gatwick at 5:25. We had an Easy Jet flight at 8:00. (Note that we knew there was another at 12:50 should we miss the intended one.) We arrived just before 6:00. As mentioned above, Norwegian gave out the passport control cards, which we had completed. We took off from the gate as quickly as we could, and got a nice, long, are you awake now walk into the border control desk. It wasn't at all busy, so that was easy. We then proceeded to baggage claim for our checked luggage. In that area there is a desk for Gatwick Connects (it's hot pink). If you have boarding passes already in hand, they will take your bags and get them to your next flight. The woman scanned our passports and boarding passes, tagged our luggage, and put it on a luggage cart/trolley. We had to walk past the camera with our luggage under our control. She then took the cart to send our luggage on, and we proceeded to security. There is no charge for this service. For a fee, they will escort you through security, which I kind of wished we had done, as it was a typical pain. Had our timing been tighter, we would definitely have paid that fee.
Now, naturally, our Easy Jet flight was at the opposite end of the airport. Another brisk walk (somewhere in there the shuttle was involved, but I don't remember exactly where that was) got us to our gate. We had ample time to use the bathrooms to wash up a bit, use the facilities, fill water bottles and get in line to board. And then that plane sat for a hour as well.
I'm not advocating such a tight schedule, it just worked out that way for us. But, perhaps some of this information will help others.