If you’re interested to learn spoken and written Newari language officially known as NepalBhasha then it’s a good news. There are few institutes that teaches Newari lipi mainly Ranjana lipi. I can’t say how many are there in Kathmandu as these institutes come and go because of less number of students and if you’re looking for spoken Newari language learning institute then I don’t think there is any institute for that. But the good news is you can find youtube channel for that. You can learn, practice from channel named “Timila Ja”. Till today they have uploaded 100+ videos and covered mostly every topic for learning language. Here you can find the channel youtube.com/c/timilaja and you can also find Newari song lyrics at nepal-bhasha.blogspot.com which you can also find it in video description.
I find these videos and blog very useful for learning Newari and hope it will also help to you too.