The answer to your question is: it depends on a lot of factors. What the weapons that they are learning how to use are, how old they are, whether they have any previous level of combat training (even if they just did recreational kickboxing workouts), who is training them, the style or styles they are learning, what the quality level of the training they are receiving is, what they are learning to do with the weapons, how hard they are working at the training versus how fast they need to be able to use those skills, the reasons behind why they need those skills, and of course who or what they are learning to fight against.
In most recreational martial arts programs and even competitive martial arts programs, it will take years for the student to become proficient in the style. The average is three to five, it can also take much longer than that to develop actual combat proficiency. There’s a difference between learning the techniques used to fight and learning to actually fight, this is part of the reason why so many people out there look down their noses at black belts. To them, what use is a black belt if the person who wears it is just going to lose out the average street thug?
Here’s what they don’t see. In most styles, the ranking system isn’t a symbol of an individual’s combat proficiency, but instead a sign of their mastery of technique. It’s a symbol of what they know and how good they are at the style they’ve learned. Now, most martial arts systems are actually older fighting forms or the revival of old fighting forms that did see military use. However, in a modern combat context they are also outdated, this means the tactics that the techniques are teaching the student to defend against (on average) are not the tactics a modern mugger or street thug will use when they are attacked on the street. This doesn’t mean that the techniques are irrelevant, it just means that they need to be modified for the situations the student will find themselves facing. Often, in order to become combat proficient, the student must find another instructor or source of information to develop that proficiency. Other, more modern, combat styles and self-defense courses are useful for learning to couple what the student already knows with what to do in the situations they will be faced with.
It’s not enough for someone to just learn how to do a technique, they must also learn what to do with it, and develop the willingness to actually do it when the necessary time arises. I, for example, didn’t learn how to stay cool in panic level situations from my martial arts experiences. I developed the talent, mostly, because I grew up in a house with parents who yelled (a lot).
Now, you may be wondering what this has to do with your character. The reason is to make sure that you choose the right path for your character that will help them be what you need them to be for your story. Military training and police training are highly intensive but only eight weeks long, a character who has a trainer who is a retired police officer or formerly in the military (or both) giving them personal instruction will reduce the amount of time it will take them to learn to fight. The trade off is that the training will be much more intensive on a personal level.
Since you didn’t mention which weapons they’d be training in, I should mention that most weapons are designed for killing. In fact, it’s very easy to kill someone with a knife or a baton. (It’s actually much easier to kill someone else period, even in hand to hand.) So, if you’re character is planning to use subdual methods, they’ll need specialized training. Someone like Michael Janich is a good resource to study up on, because the form he developed focuses on personal self-defense with a live weapon and he’s also great at communicating the concept behind the technique, which you’ll need if you’re going to try writing the technique.
You’ll also need to stop and think about what opponents she’s training to face. The time it takes to become proficient against a mugger versus the time it takes to become proficient against a professional killer (whether that be organized crime, military professionals, cops, etc) is pretty big. If she’s facing professionals, then she’ll also be at a disadvantage because in the beginning she’ll be lacking real world experience.
So, how fast will it take her to get decent? Let’s say that if she’s using a personal instructor instead of group classes and with a focus on real world combat: three months.
Good short range weapons are: escrima stick and a knife. If she’s studying one of the Filipino fighting forms like Pentjak Silat, she’ll learn how to dual wield those two weapons together. An example of a dual escrima stick user in fiction is Nightwing. The escrima stick is very effective weapon and, when using body shots, has a slightly higher margin for error. You can also conceal them easily beneath a coat and, when using bamboo ones, sneak them past metal detectors.
So, it’s worth thinking about.