There was some comment in the meta thread to the effect that answers from non-experts are useless.
Nevertheless, I thought to indulge myself with a few remarks (that actually address only a small part of the question). The objective is to get experts to correct me,
if they can be troubled to do so. With such a contribution, perhaps this answer can be useful to others as well.
My answer to the original question is that there are no substantial relations at this point. This is because
mathematical logic is concerned mainly with mathematics;
philosophical logic is concerned mainly with philosophy.
This characterization is obviously an oversimplification, but I wonder if it does not capture the
distinction in all essentials.
Here is a short Wikipedia description of the research of A. C. Grayling, who, I gather,
is a rather distinguished person in philosophical logic:
`His principal interests in technical philosophy lie at the intersection of theory of knowledge, metaphysics, and philosophical logic, through which he attempts to define the relationship between mind and world, thereby challenging philosophical scepticism. Grayling uses philosophical logic to counter the arguments of the sceptic in order to try to shed light on the traditional ideas of the realism debate and developing associated views on truth and meaning.'
This sounds like philosophy to me.
On the other hand, if we examine the work of leading people in model theory (the only branch of mathematical logic with which I have some passing
acquaintance) like Udi Hrushovski, Angus Macintyre, Anand Pillay, and Boris Zilber,
it is hard not to think it looks like `generalized algebraic geometry.' Indeed, applications to algebraic geometry and number theory
form a mainstay of their work.
As to the reasons, I gathered some insight from some amusing passages in the preface of Van Dalen's textbook on logic (which I do not have on hand right now).
He writes of the 'sacred' tradition of mathematical logic closely related to Hilbert's programme and the incompleteness phenomena, where
foundations were handled with great care and awe. He then goes on to describe his own encounter with recursion theory lectures
by Hartley Rogers, where logic was treated like any other branch of mathematics, say linear algebra or complex analysis.
This he refers to as the 'profane' tradition, obviously more distant from philosophical origins. I wonder if it isn't the case that mathematical logicians simply became bored with
the sacred tradition (in keeping with the twentieth century trend to find many sacred things boring).
In any case, it seems relatively clear that the profane tradition is more dominant among current day practitioners.
One way to see this, according to an old conversation with Hrushovski, is that papers in mathematical logic contain as many mistakes as those in algebraic geometry.
Because of my ignorance, I may be missing the possibility that Proof Theory and Set Theory are still somewhat close to philosophy.
But the simple-minded answer still seems to be a reasonable one.
Just after posting the above, I noticed an obvious flaw in my own argument. I could have written, for example,
mathematical gauge theory is concerned mainly with mathematics;
physical gauge theory is concerned mainly with physics.
But it would be ridiculous to claim that there are no substantial relations between the two. So if my conclusion is correct, it would require a more elaborate discussion. Oh well, perhaps later.
Here is a response to my own objection. The difference between the two cases mentioned above has little to do with logic and gauge theory in particular. That is, mathematical logic and philosophical logic have little in common simply because mathematics and philosophy have little in common. Therefore, difference of purpose is enough to produce a divergence in methods and ideas. Physical problems, on the other hand, are resolved in the language of mathematics. Hence, commonality of origin becomes enough to maintain a tight thread between, say, the two gauge theories.
This superficial analysis is all I have time for now, but maybe it's plausible.