I used to teach English as a foreign language to Arabic speaking students (children and adults). I also did a ton of research into teaching methods that are appropriate for adult learners who are unfamiliar with the Latin alphabet, and not only taught reading but also wrote courses in it and trained other teachers to teach it. I'm also dyslexic and learned how to read and write in Arabic as an adult (the breadth of my Arabic knowledge is narrow but I can read and write it, and understand what I read as long as it's words within my narrow vocabulary). So I have personal experience of doing this (both as teacher and learner).
How long it takes really depends on the individual. Fast learners with good visual memories will learn to read quickly (like in a matter of weeks). Slower learners will take longer. People with dyslexic tendencies or dyslexia will really struggle. Synthetic phonics is usually the best method for people with dyslexia/dyslexic tendancies (due to lacking the visual memory for remembering so many words by shape*) as it relies more on logic. For story purposes, if you need your character to pick it up quickly, then have other ways to show your character is a fast learner (maybe they were always top of the class at school or something). If you need your character to struggle, then maybe show they have dyslexic tendencies (e.g. tendency to confuse left/right, clockwise/anticlockwise, maybe they were one of the last in their class to get the hang of learning to read in their own language. Maybe they know they're dyslexic or have dyslexic tendencies.)
*there are different kinds of dyslexia and it's not necessarily about the strength of visual memory, it can be the way the visual memory works. For example was and saw are the same basic shape, only inverted. Begin able to remember the shape isn't enough - you have to also remember which way round the shapes go.
So anyway, back to the other part of your question about whether the learning time would be shorter if they can already read in another alphabet. The answer to that is "it depends". This is because in English and similar alphabets, words are broken down into sounds (phonemes) which are usually represented by one letter, but in English (just to be awkward) can be represented by two or three (occasionally more). For example, cat is c/a/t. Three is th/r/ee and chair is ch/air (or ch/air/r in American English). If you're going from an alphabet that works like this to another that works in the same basic way, then yes already knowing how reading works would mean it's a lot quicker to learn. However, not all languages write in this way. In Arabic, for example, words are broken down into slightly larger units. The name "Muhammad" in Arabic is written in four letters (I'll use English ones to represent them), i.e. M H M D and short vowels go with these letters. Like Mu/Ha/MMa/D -four letters. It can be a bit tricky for beginners to grasp why in English, Muhammad has to be spelled with so many letters when you've only ever experienced an Arabic style alphabet where it can be expressed perfectly well in four letters. Being able to break it down into separate consonant and vowel sounds (when you're an adult and never had to do this before) is what makes it tricky, i.e. going from Mu/Ha/MMa/D to M/u/h/a/mm/a/d. This would mean it takes longer than someone who's first alphabet is more like English, but would still be quite a bit quicker than for someone who's not literate in any language.
From the research that I did before starting teaching all this, languages where the written language is broken into bigger units than Arabic, e.g. symbols representing syllables or even entire words, has the same issue I've described as with Arabic, only that it may be even harder for an adult learner to learn how to break syllables down into individual sounds. I don't have personal experience of this. I'm only going by what I've read.
Also bear in mind that students who are fast learners with strong visual memories may not need to learn how to break down words at all, because they would just remember what the words look like - the disadvantage is not being able to read unfamiliar words so all students benefit from learning how the system works.
So the answer to that really depends on what the character's native language's writing system is like.
There are lots of other variables too... a bad teacher will slow the learner's pace of learning right down. Especially if the teacher is relying on look-and-say methods and the student has a weak visual memory and dyslexic tendencies. Also, things like how long you spend practicing will make a huge difference. Practicing once a week probably isn't enough. Practicing several hours a day will mean you learn it all much quicker.
So really you could make your character learn English literacy as quickly or as slowly as you like, as long as the speed matches the variables. You're looking at some students learning in a few weeks versus others maybe still struggling after a couple of years, if none of the factors have fallen in their favour. Or possibly giving up altogether.
If your character can already speak English, that will go in his favour as he'll be able to recognise when he's reading words correctly and be better at using contextual clues to know what word it is, e.g. "I had a row with my partner" or "I like to row my boat on the lake" - someone whose English vocabulary isn't strong wouldn't know which pronunciation of "row" to use in those sentences, but someone with a good vocabulary/understanding will know from the context. Your character's learning won't be limited by their lack of knowledge of English words. Many of the students I taught were beginners in English, i.e. very little vocab or reading skills. I taught some students who were pretty fluent in listening/speaking but due to dyslexic tendencies really struggled with the literacy side of it.
Hope that helps.