The best source for information on this topic would be Chapter 14 ‘Nuclear Destroyers and Frigates’ of ‘U.S. Destroyers: An Illustrated Design History’ by Norman Friedman. This book is on Google Books but I’m not sure if this chapter is online content. Check it out.
To summarise there were plans in the late 50s for three nuclear escorts, a light cruiser sized “frigate” (DLGN), a destroyer (DDN) and a nuclear powered ASW destroyer escort (DE). There was one 1957 requirement for 16 DLGNs to provide escort to six CVAN and six CGN when the US Navy planned to build one nuclear carrier per year. In the end budgets played their role and only two DLGNs were built *.
The operational deployment of a nuclear CVBG would be very similar to conventional one except they could keep a constant 30 knot speed to make it very difficult for submarine attack. Conventional CVBGs would only sail this fast within 500 NM of the target area because of fuel consumption. The destroyer was judged by the post WW2 experts as being the one surface ship in most need of nuclear power, thanks to their high reliance on refuelling, but it wasn’t to be.
What sunk the plans for a fleet of DDNs to escort the CVANs were the nuclear reactors increased cost and complexity and most significantly their lack of power to weight ratios. The late 1950s design for a 4,300 ton (3,500 light) DDN with two S4G reactors and the weaponry of a Forest Sherman could only make a top speed of 27 knots. Replacing one reactor with six 7,000 shp gas turbines enabled a dash speed of 30 knots (22 knots sustained) but only for 33 hours. Without a more powerful lightweight reactor nuclear power was not an option for a 30 knot ship under ~6-7,000 tons.
These were only sketch designs and if there are layout drawings of them they haven’t surfaced from the NAVSEA archive. You could build a nuclear powered destroyer with the combat systems of an Arleigh Burke class today but the cost driven defence budgeting would kill it well before it got close to a shipyard.
* These were the US Ships Bainbridge and Truxtun. Despite many published sources claiming they were nuclear powered ‘one offs’ or ‘versions’ of the Leahy and Belknap DLG classes respectively nothing could be further from the truth. The Bainbridge was an original class designed around nuclear power (you can’t just pull out steam boilers and replace them with nuclear ones) and of course took on as much of contemporary combat systems that could fit its space and weight margins which was the same as the Leahy. The Truxtun was a repeat Bainbridge but with a SQS-26 bow sonar. This consumed so much space and weight that a forward mounted missile system like a Leahy could not be accommodated. So it was built with a one end missile system (aft) and a 5” gun occupying some of the unconsumed bow space and weight. Any resemblance to a mirror image Belknap was purely coincidental. I’ve even seen it written that the Truxtun was a nuclear Belknap with the armament reversed to improve the firing arc of the 5” gun! Complete fantasy based on a simplistic reading of the cosmetic appearance of the ship. It beggars belief that such a hollow analysis can pass for ‘respected opinion’ on such matters when research of the original materials like that by Friedman is available...