Not needing to reclaim baggage obviously improves your chance of making the connection, but has no effect on waiting time at Immigration, since as in most places baggage reclaim is after Immigration. The personal information you've given suggests that once you actually get to see an immigration officer you'll be on your way very quickly indeed. The problem, which may or not exist on the day, is the number of people ahead of you in line. All depends on which planes arrive in which order, and how many of their passengers need to go through the non-EU/EEA line with you. Then, if everyone in front of you is as straightforward a case as you are, the line moves very fast. As soon as some of the positions get blocked by people whose documents are questionable or are trying to sidestep the regulations, things slow down dramatically.
Once you are past immigration, walk through baggage reclaim and out through the Blue (arrivals from EU) exit. You're most unlikely to be stopped by Customs, and if you are it will be a very quick question.
Then you're landside and need to find your transport to Gatwick.