PhD in autonomous underwater robots
This is a very applied field, I don't think there are PhD programs related to directly to underwater robots. I did my PhD in this exact field :) so most of this is my experience and knowledge of the field.

Generally you shouldn't worry much about what the PhD program says exactly, as this field is very application driven, so Autonomous Underwater Vehicles (AUVs) are usually designed and built in places where they are needed, close to shore or to oil and gas industries. This usually relates to PhD programs in marine or oceanographic sciences, but the most important thing is the advisor, who can guide and support you in your research.

In a University with "application" departments (like marine science) and "basic science" departments (like CS and Engineering), you can do a PhD in Science or Engineering of AUVs while working to solve some problem related to the application side.

A quick way is to go to Universities and Research Centers in Robotics, for which there are many and they usually cover application fields like underwater, space, rescue, etc. For example, Carnegie Mellon Uni has a Robotics PhD program, for which there should be Professors interested in Underwater Robotics. As you mentioned MIT also has a partnership with WHOI and they do a lot of Underwater Robotics.

But as you mentioned, not many Universities touch this subject. In general Research centers (not always affiliated to a University) usually have the lead in this topic.

In Europe, the leader is definitely the University of Girona with its CIRS research center. They design/build AUVs, and develop algorithms for autonomy. The German Research Center for Artificial Intelligence, Bremen branch (where I work) also builds AUVs and algorithms for autonomy.

In the UK there is also the National Oceanography Centre in Southampton which does a lot of Marine Science supported by AUVs, so they also design/build AUVs and related algorithms. In Norway's NTNU there is AMOS.

I know that there are AUV many researchers in Asia, like at the University of Tokyo and the Tokyo University of Marine Science and Technology.

This list is not exhaustive, AUVs are developed all around the world. About PhD positions, that is something that depends on funding, time and topics, so its up to you to search these. Hope this information helps.