Considering a second PhD, this time in statistics. Will any professors be interested in working with me?

As most of the comments already say, doing a second PhD is probably not a good use of your time and not very helpful. From an industry perspective you have a math PhD, that means you are very smart and can learn anything, especially any maths very quickly. Learning some statistics or machine learning can be very helpful for an industry career but there is no need to get a formal degree in it. You can state in job applications that you have a math PhD and that you know statistics. Even if the statistics is entirely self taught in the real industry world this is equivalent to a PhD in statistics.

From your question and comments it seems that there are two distinct issues:

• Finding a good industry job with your profile: since you mention statistics I assume that you are looking at jobs in the area of data science. There is so much demand in this area that you don't even need a formal diploma, you could just teach yourself and acquire some experience with online material. In case you really want a diploma, what you need is a Master, not a PhD.
• Keeping doing research because you enjoy it: you don't need to do another PhD in order to keep practicing research as a hobby. Most advisors would be reluctant to take you on anyway if you are not able to work full time on the PhD: they would often see you as a "flight risk", since you don't have any specific funding and no strong reason to finish a (second) PhD. However you could certainly collaborate on some research projects, many academics would appreciate a skilled collaborator that they don't have to pay.

I'm a retired Uni academic.

If you are determined to study more (and bravo for that), I would strongly recommend an MSc or an MMath in the subject. If you are short on skills, you will get far more from a good Masters. It will be cheaper, quicker and more intensive. Frankly, you will learn more as well. You've proved you can do research and that's the end of PhD for you. You would slow down your career too much by doing another.

Distance learning is appropriate if you are strapped for time.