I've done ballroom dancing and many martial arts. While I did find some things that I could take from one to apply to the other, generally I concluded that they're fundamentally different. Let me explain.
In ballroom dancing, a male lead needs to be able to signal to his partner what he's about to do. The signal comes from a solid frame that provides pressure that ones partner can pick up on and use to know what they're supposed to do next. The goal is to communicate intent in order to synchronize movement between two people. One is the leader. The other is the follower. And the follower needs to understand what the leader wants to do, so the communication from the leader of his intent must be strong and unambiguous.
In (most*) martial arts, on the other hand, you have two competing individuals. Their goals are the exact opposite of that of dancing. They are trying their best to hide their intent from their opponent, unlike dancing which tries to make it clear and obvious.
In both martial arts and dancing, timing is indeed everything. But instead of working with your opponent's timing and synchronizing each others movements, the opposite is required in martial arts. You're trying to disrupt your opponent's rhythm or use it against him.
So instead of your opponent doing something, then you do something, and so on back and forth, the rhythm in martial arts is such that if your opponent begins to punch, just as he starts, you begin your counter attack with a step to the side and a strike of your own that's timed to land before he completes his strike. That's called "off-beat timing" or "half-beat timing". In other words, it's working against the rhythm, not with it.
In martial arts, you don't want to get in the mode of your opponent doing something, then you do something, then he does something, and so on. That's wrong. You want to become unpredictable and time your techniques in such a way that causes him to have to stop what he's doing and recalculate. Never become predictable or telegraph what you intend to do.
*I said "most" martial arts. There are some that don't do competition at all (aikido for example), where the goal is to work with each other, blending in and working with the rhythm of your opponent. There are some that are performance based (contemporary wushu for example), where the goal is to sort of do a hybrid of martial arts and dance. In these cases, ballroom dance training can actually have a positive influence.
Now that being said, I have heard on multiple occasions dance instructors saying that their students who came from a martial arts background were generally better. And their reasons have to do with being able to understand body movement in general. Martial artists are more aware of what their bodies are doing, spatially. They're more aware of their "frame" and whether their partner is getting a strong or weak connection to it. They understand the importance of keeping their form and not forming bad habits like looking down at the dance floor or slouching their shoulders. They come in more disciplined, more focused, and ready to learn. Stuff like that.
The biggest problem for me coming from an extensive martial arts background into beginning ballroom dance classes was the way dance is taught compared with the way martial arts is taught. That's going to cause martial artists the most frustration, I think. It did for me, anyway.
In martial arts, you're typically given a partner who already knows what to do and can teach you. If you're struggling, the instructor comes around and makes sure you know what you're doing. You get continuously corrected, whether you want it or not. Very rarely do you ever feel like you don't know what's going on.
In dance class, your partners are the followers, and you're a leader. (Or vice-versa.) They know their part, but they can't tell you your part. They don't know that. So you're on your own. The instructor will demonstrate something for the whole class, and then you just have to remember it all from memory and not screw up. It's especially important if you're a leader, because leaders need to start each movement and signal it to their partner unambiguously. If you're having trouble with it, or if you can't remember what to do, you literally can't continue. You just have to stand there and hope the instructor will be around to fix you. But the instructors don't typically make the rounds to correct you. You can raise your hand and hope the instructor will come to you to help you out, but that's not guaranteed. And if you don't get correction on this step you're learning now, you can't continue with the rest of the class, because the next step you learn builds on the current step.
It is so inferior to the way martial arts classes are taught that you'll wonder how anyone ever learns dance this way.
I later realized that "group" dance classes were only there to practice what you already know, with different partners that rotate from one person to the next. Some/many people actually do learn it that way, but pretty much everyone would be far better off booking time with a private dance instructor instead in order to learn it. Then when you're ready, you can do the group classes and social dances to get better at it with lots of other partners. Then you find out for real if you know it or not.
Contrast that with martial arts classes. We don't need private lessons at all, for the most part. The regular classes go over everything really well. You're going to get correction and feedback all the time. And your partners can teach you, even if they themselves just learned it.
Anyway, I could go on about the differences between dance and martial arts. But suffice it to say, there's not a lot that I think one can benefit from the other. My best advice is to go into each with an empty cup. You'll be better off in the long run. And do get a private dance instructor at first. You'll save yourself months of agony and frustration. My advice for what it's worth.
Hope that helps.