The crew do not want to die, nor do they want to be forced down and captured. So they decide to send out a message to any US or Russian radios that are receiving them that the intrusion was an accident, that they will be reversing course and heading back to the arctic circle, and if they are attacked or if US planes approach too closely, they will in no uncertain terms, deliver the bomber's payload to the ground beneath them.
Everything about your scenario is likely, and probably happened to varying degrees, except that last part. Escorting an intruding aircraft out of your airspace is standard procedure.
The Soviet crew would have been trained on this eventuality and expected this. There'd be no panicked argument among the crew, that's Hollywood, nuclear bomber crews are professionals. They train again and again and again until every emergency is normal and they can handle it almost literally without thinking. What to do if you blunder into enemy airspace would be one of those emergencies they'd be trained on. They'd follow procedure.
Getting an escort reduces tensions by allowing the defending nation to have eyes on the intruding aircraft and know it's taken no hostile action like opening its bomb bay doors, turning, maneuvering aggressively, or using electronic counter-measures. For the crew, who has just made a gross navigation error, it gives them someone to communicate with, even if just with hand signals and lights, and follow out of harms way.
"Russians don't take a dump, son, without a plan". Threatening to drop their bombs without orders to do so is the last thing a Soviet bomber crew would think to do. A bomber pilot would not have the authority to decide to drop their bombs without orders, nor are they trained to act independently. Taking such a reckless and independent act would be the antithesis of Soviet military and social training. Unlike the US, the Soviet military was tightly controlled. Plans are big and rigid. Junior officers are expected to follow orders to the letter.
So that part of your scenario is highly unlikely. Instead, they would follow procedure. That procedure would probably say to declare an air emergency on normal international emergency frequencies and request escort out of US airspace. They'd accept peaceful escort out of US airspace, face discipline when they got home, and the diplomats would deal with the (fortunately figurative) fallout.
Keep in mind, the Soviets are likely flying the Tu-95 "Bear", a four engined prop driven bomber roughly analogous to the US B-52. Very, very long ranged with a large payload. While it's much faster than you'd think, it's not supersonic, and it has very little hope of fighting or evading its way out of US airspace.
Like the B-52, the Tu-95 soldiers on while sleeker, faster bombers fall by the way side. While the Soviets had supersonic bombers, notably the Tu-22M "Backfire" analogous to the US's B-1, they did not have the range to casually blunder into the US.
What is the most realistic US & Soviet response to this anomaly?
"Oh shit, we have a panicked or traitorous crew of a nuclear bomber who might start WWIII!" Both sides would think this, for the reasons stated above. There's no reason for them to be so terrified of being shot down that they'd threaten to bomb the US if approached. This bomber crew has gone rogue.
The Soviets would check the background, mental state, and political allegiances of the crew to find a motivation for their going rogue. Is panic cover for an act of treason? Did they go mad? Are they exhausted and doped up on uppers to stay awake? Perhaps they've been overworked and have been abusing stimulants?
One plausible scenario is that a portion of the crew has been incapacitated resulting in the rest being overloaded and unable to catch any sleep. Maybe the commander and navigator are out. The co-pilot and others have taken over their duties. Overworked, stressed, and exhausted deep in enemy territory with their minds jumbled by amphetamines, they begin making bad decisions.
"How the hell did they get there?" With a cruise speed of about 700 km/h, it would have to be a hell of a navigation blunder for a Tu-95 to wind up deep in US airspace without noticing. That's hours flying off course. Highly unlikely, even a simple look at the Sun would let them know they're going the wrong way. Both sides would have great suspicion about the crew's claim of a navigation error which would add to tensions. The crew's threat would not help matters.
The US would wonder "how the hell did they get there without us noticing?" As mentioned above, the Tu-95 is not exactly a sleek aircraft, and to have the range to reach the US it would have to be flying slow and high over the poles. Earlier in the Cold War, US detection systems were full of holes, but later and later they improved. Still, the US early warning system had flaws and things did blunder through. The US would wonder if this was a test of US early warning systems adding to tensions.
The US and Soviets would scramble to find out if the bomber crew can follow onto their threat. The US would be asking the Soviets if bomber crew could arm their weapons without explicit orders and codes. The Soviets would be verifying that. I don't know what Soviet bomber procedure was, and neither did the US.
If the Soviets can confirm the bomber crew cannot arm their bombs in flight without orders, the US would likely tell the bomber crew to accept an escort, and take no evasive maneuvers. They'd tell their interceptor pilots to shoot the bomber down upon any hostile act such as evasive maneuvers, changing course, or opening the bomb bay doors.
There would be hawks in the US arguing to force the bomber to land so they can impound it and its crew. US military intelligence would be drooling at the opportunity to interrogate a Soviet bomber crew and pick over a Soviet nuclear bomber complete with nuclear weapons. Some would be pouring over international law and treaties to build up a legal rationale or loophole.
Meanwhile, others would think "how do we talk these guys down?" US diplomats would get on the phone to Soviet diplomats and try to patch through Soviet officers to speak with the bomber crew. If they can calm the crew down, they'd then be escorted out of US airspace and into the loving arms of the Soviet military police.
If they can't talk them down, it's likely the US would send in escorts in, whether the Soviets liked it or not, whatever threats the Soviets are making. They may wait until the bomber is over sparsely populated territory before approaching. If the bomb bay doors twitch, or if the bomber takes any evasive action, their orders would be to shoot it down and hope the crew did not have time to fully arm the bomb.
Would the outcome change if this happened later in the Cold War?
It would only make it better. As the Cold War progressed the speed and intensity of nuclear war increased. As more incidents occurred, the danger of accidental nuclear war became apparent, and more and more safeguards were put into place.
The later in the Cold War this occurred, the would be easier and faster for the US to contact high Soviet officials to get a confirmation that this is not a sneak attack. Prior to 1963 it would have to go through normal diplomatic channels which could take some time.
The Moscow-Washington hotline was established in 1963. Far from being the iconic "red phone" it was a teletype machine (basically a telegraph line hooked up to a printer and keyboard), then in 1986 a fax machine (basically a printer hooked up to a phone line), and finally in 2008 a secure computer link via email.
Later on in the Cold War, nuclear bombers would be increasingly replaced with missiles. First and retaliatory strike capability would increasingly move to ballistic missile submarines. Your scenario would probably shift to a rogue nuclear ballistic missile submarine that finds itself damaged deep in Soviet territory.
Finally, radar technology would get better and better, as would radios, navigation, and interceptors. It would become increasingly unlikely the crew would get accidentally lost, and that nobody would notice. This would serve to increase US suspicions that this "lost bomber" is cover for testing a new way to penetrate the US early warning system.
A variation on this scenario is famously depicted for comedic effect in Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb which if you haven't seen you really should.
Dr. Strangelove, a comedy, was based on the book Red Alert (aka Two Hours To Doom) which was not. Red Alert, in 1958, outlined a scenario very similar to your own.
A US Air Force general considers war with the Soviets inevitable and decides a preemptive attack is the only option. He uses a procedure designed to give him authority in case the US government is knocked out in a first strike to order his bomber wing to attack. The book is about the cascading consequences of procedures designed to expedite nuclear attack.
Operation Chrome Dome, the US operation keeping nuclear bombers continuously in the air from 1960 to 1968 to keep a credible second strike capability ready, made these scenarios even more likely.
Trinity's Child, and the movie adaptation By Dawn's Early Light, are another similar similar scenario. The book and the movie differ in their details, but both depict attempts to stop a nuclear war and recall bombers.
The film Crimson Tide is another film with a similar scenario, but this time it's a US ballistic nuclear submarine.
A US submarine captain gets a partial transmission that looks like a launch order, but is attacked and forced to dive. Its radios are damaged. The captain believes nuclear war has begun and wants to launch. His XO (second in command) wants to wait until the radio is repaired to verify the order. An escalating fight for control of the sub ensues.