Go out to a guitar or ukulele forum right now and post this question. Wait about an hour and look at the responses... I'm betting that you have just started a forum war. Half with say you should, while the other will say you shouldn't. Both opinions will likely not answer the question for you. And, at the end of the day, you will still struggle with the question in your own head.
So, here's my take:
I've never met a musician who learned to read and write music who said it was a waste of time. This is true of all instrumentalists: guitar, ukulele, piano, trumpet, accordion, etc. It has always been a rewarding pursuit.
That said, to be proficient at it, you will have a lot of work ahead of you.
Why it's a good idea:
Basically, it boils down to communication. If you cannot effectively communicate your musical ideas to other musicians (or to your future self), then you are setting your music up to fail. reading and writing music is how we communicate "off instrument". Words don't work for us.
Think about it like music is a foreign language, like Italian. You can listen to tapes and learn to speak the Italian, but as soon as you are reading the menu at a cafe in Venice you are going to be at a loss because you never learned how to read those words properly. Sure, you might stumble through it, but there will be a loss in translation.
Music is no different. You can listen and learn how to play a tune, but as soon as you are trying to get your point across to other musicians off instrument, you're lost.
Amateur vs. Professional
Obviously, in the guitar and ukulele world (and many other fretted instruments) there is clearly no emphasis on reading music efficiently. We are only really expected to read chord symbols and lyrics off of a chord sheet and that's about it. And that is totally fine. There is NOTHING wrong with being an amateur musician. Keep it simple and keep it fun. That's the point!
But, if you are a budding professional musician, then you need to dedicate yourself to understanding the craft. Sure, you can play "All the Things You Are" with a beautifully crafted chord melody and bass line. However, can you communicate what you are doing so the rest of your band can understand where to come in?
Maybe you intend on being a soloist your entire musical career... That's fine too. Nothing wrong with that. But, as many musicians will agree, your future self is very forgetful. Having a way to document how you did something makes remembering it later as easy as reading.
In the long term, learning to read and write music is a great skill. If you only get into it a little as an amateur musician, that's fine. But, as you work your way towards being a musician of professional quality, it is essential that you can communicate with other musicians.
~Danny
P.S.: There is another reason you should learn to read music: money. There are literally thousands of musicians out there that are trying to get noticed. Being able to read off of a piece of sheet music will get you noticed. Getting noticed gets you hired! I've had a few jobs that came to me because I could read and the other guy couldn't (these were musicals). It is worth it!
