Don't confuse immigration and customs. Yes, there are both between Italy and the U.K. as the U.K. has not acceded to the Treaty of Schengen (customs/immigration union).

You will have to clear Italian immigration at FCO (or Ciampano) - they'll most likely just stamp your passport unless you have a passport or have an I.D. card of a E.U. member nation - and go through customs though the latter is usually a mere formality of walking through (go through the E.U. channel - you'd most likely not see a soul there).

As for how long it takes to go through, maybe as little as 5 minutes. Or maybe longer, depending on whether you're bussed in from a hardstand and/or if there's a lot of pax waiting to clear immigration ahead of you.