The tier of generals just not senior enough to have a seat on the doomsday planes isn't in the emergency line of command to the nuclear weapons. So, regardless of how powerful a small coalition of those generals is, they cannot reliably prevent a nuclear launch. (They'd need a pre-existing conspiracy to quickly and efficiently turn their own air defence batteries against their own doomsday planes... at which point it seems very likely they'd just launch a coup long before a nuclear strike was ordered.)

So, I guess our last hope is that a small conspiracy of generals just under the doomsday plane tier would stage a coup once the nuclear sabre rattling reached a sufficient magnitude, before the nuclear first strike order is given.