He doesn't mention it above, but another plane flying in formation with the Enola Gay dropped measuring instruments by parachute. I found aversion of the story in the New Yorker that incorrectly describes this device like a timer. "...wires on top of the device were attached to a solenoid unit on the roof of the bomb bay. When the bomb dropped out, the wires came loose from switches inside the clock-box—the brain that told the bomb to drop for forty-five seconds before detonating." It's correct that a solenoid switch started the sequence, but the count down clock was a back up system. More here."...Weak radio signals were being pumped down from the bomb to the Shina Hospital directly below. Some of the radio signals were absorbed in the hospitals walls, but most were bounced back skyward. Sticking out of the bomb's back, near the spinning fins, were a number of whiplike thin radio antennae. Those collected the returning radio signals, and used the time lag each took to return as a way of measuring the height remaining to the ground. At 19,000 feet the last rebounded radio signal arrived."
The radio described by Bodanis was the AN/APS-13 radar unit. While I can't find a source on the frequencies used by the equipment in this case I do know that a standard AN/APS-13 operates at 410 - 420 MHz with a receiver IF of 30 MHz. The British called these "Archies." One source I read claimed that it detonation was triggered only when two AN/APS-13 radar units identified the critical altitude. They were trying to prevent premature detonation. With the score of the number of other back up systems that seems totally plausible. Some sources claim there were as many as four. More here. They were used as tail radar in allied planes in WWII. This is the device that actually triggered the altimeter fuse. The Duxford Radio Society has pictures of a restored unit here. The device is automated and self contained.
Some of this is still classified today, and I suspect this was not an off-the-shelf AN/APS-13. That device wasn't designed to measure distance. It was a tail warning device, i.e. it warned a pilot that an aircraft was approaching him from the rear. It's effective range was given as 2,000 to 2,500 feet transmitting and receiving a fan-shaped beam behind the airplane. More here. You know the rest of the story but let me quote this from the Yale Avalon Project, and recommend that you visit their website.
"At 8:16 A.M., the Tokyo control operator of the Japanese Broadcasting Corporation noticed that the Hiroshima station had gone off the air. He tried to use another telephone line to reestablish his program, but it too had failed. About twenty minutes later the Tokyo railroad telegraph center realized that the main line telegraph had stopped working just north of Hiroshima. From some small railway stops within ten miles of the city there came unofficial and confused reports of a terrible explosion in Hiroshima. All these reports were transmitted to the Headquarters of the Japanese General Staff."