The answer is "Not at all." The problem of other minds is a metaphysical and epistemological question: "How do I know that there are other minds?" I can directly observe behavior of other humans, but to infer mentality from behavior in the case of others is to make an inference, and therefore the question arises whether the inference is a good one.
The question about The Other, so far as I can tell, is ethical. "What should I do to/for/with the Other?" Of course, the complicating factor is that sometimes authors like Levinas tend to make (seemingly implausible) claims about The Other that look metaphysical, like, "You can't be a self until you encounter an Other." I am not qualified to speak to what the reasons for such claims might be, however.
Let me expand upon why I think the metaphysical claims that the encounter with the Other being a transcendental condition of the possibility of your own subjecthood is implausible. Doing so might help explain why analytic philosophers aren't inclined to think "the Other", as Lacan and Levinas describe it, is in fact a useful category at all.
Here's the argument:
(1) If encountering an other is a condition of the possibility of subjecthood, then you can't be a person without encountering the other.
(2) But you can be a person without encountering the other.
(3) Therefore, encountering the other is not a condition of the possibility of subject hood.
(1) is just a definition that expresses the meaning of the phrase "condition of the possibility".
We can support (2) in two different ways. First, by thinking of the case of severely autistic people. Autism is a spectrum disorder that involves impairment to one 'theory of mind' i.e. with one's ability to recognize and perceive that other people are people, as opposed to mere objects. At the far end of that spectrum, it looks like we have individuals who genuinely could not ever "encounter an other". Such individuals tend to suffer a number of really heart-breaking impairments, but it is obviously absurd to say that they aren't subjects at all.
The second way that we could support (2) would be to appeal to conceivability/possibility arguments. Suppose you don't buy the autism case in the case of humans. Ok. If it is conceivable without contradiction that there exist any kind of subject (aliens, God, whatever) that does not encounter an other, then it is possible to be a subject without encountering an other, just as (2) says.
(3) follows from (1) and (2) by modus ponens, so the argument is valid. It looks sound to me as well.
I'll admit that I've never seen anybody explicitly run this argument in print, but I am quite certain it's the first thing that an analytically trained philosopher of mind reading Levinas is going to think.
I would also be surprised if any argument like this had ever been published in the Levinas literature either---continental papers don't tend to criticize the great names in the canon, or object to their theses, or try to defend those theses from the objections of others. I'd be happy for somebody cite me an article to show I'm wrong in this case.