These questions are getting hard to answer because part of the answer depends on your motivation for gaining any of these skills.
Although some (many) machinist's mates would consider ELT to be a collateral duty, the duties as a diver or a welder are even more so because they are rarely used. Most ELTs use their training and specialized skills every day. The jobs performed by ELTs are an important part of the periodic reactor safety examinations conducted on each nuclear propulsion plant and any boat or ship that treats ELT as a collateral duty and does not allow them to give the proper attention to their duties is likely to see the consequences show up during one of these examinations.
If nuclear MMs are still receiving welder training after prototype (again, I read on these boards that this training is no longer being given these days, a fact I cannot confirm or deny), I would guess that this is in smaller numbers than for ELTs. In my experience, a submarine has only one or two of these welders.
I would say it is most difficult to become a diver. First, the submarine must send the candidates to diver school. This is a significant period of time that a person is away from the ship, and one factor the ship will consider is the manning level of each division.
Second, I believe that each ship is required to have at least two divers on board. That means if there are only two divers, they are not going to get a break from going to sea and, depending on the operational status of the ship, may have a hard time getting leave. While most ships might have three or more divers, they are selected from the entire crew, not just the nuclear trained community.
A ship is not likely to select an individual for training as a diver unless that person has completed all watchstation and warfare qualifications.
While Beer Court mentioned that in his experience he has seen one welder/diver and three ELT/divers, in my experience (3 submarines) I recall no ELT/divers, no welder/divers, one or two MM/divers, one EM/diver, and two RO/divers.
Since diving is an additional, hazardous duty, the ship's divers receive extra pay. However, if the ship has more than the alloted number of divers (I think the number is three), any additional Navy diver trained individuals do not get extra pay.
Please research the physical requirements to qualify for diver duty before including it on your wish list. The ship has absolutely no incentive to request a waiver for individuals who do not meet these minimum requirements, even if such a waiver could be granted.
Again, it is the ship that decides when and if to accept volunteers for diver duty, based on a need to have a certain number of trained individuals. As with most things, a request for diver training is usually handled by special request chit, and must be approved by the chain of command. In my experience, if a ship decides they need an additional diver or divers, they first solicit volunteers. Then there is a "try out" to see which individuals can pass the minimum physical performance requirements and the medical requirements. Out of that pool, the ship selects one or two individuals to attend dive school.
