I'm surprised it wasn't picked up earlier to be honest. Anyway, sorry, this reply will probably be hideously long. My non-id twin boys have nystagmus detected at 7 weeks old by the health visitor. They went on to have developmental delay and also have other problems.
After various tests, they still only THINK it's due to a condition called ocular albinism. I want to stress their developmental delay and learning difficulties have nothing to do with the nystagmus - it's a totally separate thing. I'm sure you'd know by now if your DS wasn't meeting the usual milestones and I hope all is well for you.
Anyhow, because of their related problems, my boys have always been in a SEN school but I think I can help with some of your questions. Whether the school class your DS as having SEN will depend on the level of eyesight he has and if any adaptations have to be made. They will make adjustments for him (or they absolutely should) if they are recommended by a specialist/the Visual Impairment Team/Opthamologist/Senco.
The diagnostic journey can be quite long. First they took a developmental history of myself and (now) ex-h, then tried a few tests at the Eye Hospital but really, the 'good' tests couldn't be done for a good few years until the boys could co-operate. Some of them involved watching a series of patterns in a dark room with electrodes stuck to their heads so a wriggly toddler doesn't want to do that!
Learning to read should be ok as books have such a large print for young ones, it's not a problem. In later years, when fonts are smaller, magnifiers are very useful and sometimes a reading and writing slope. Computer fonts are easily made bigger and screen work enlarged. Having to sit near the front of the class is generally a must but again, depends on the vision he has. Sharing books can be difficult with nystagmus as many children have a 'null' point - a way of tilting their head or looking at an item in a certain way which will minimise the wobble and help them to focus.
My boys prefer to just hold things very very close. They both sit in front of the TV, directly in front of it! This isn't harmful and I was told not to discourage it.
Some people with nystagmus have quite good vision, some don't. My boys are registered as partially sighted as they need things to be 6 x bigger than someone with 'normal' sight before they can see it clearly. They manage ok. They are clumsy though. They find sports difficult (a thrown ball is hard for them to track for example). They won't be able to drive.
But, many people with nystagmus aren't classed as partially sighted and just need a little help - bigger fonts now and then, etc.
I hope I've not worried you. People rarely seem to notice my boys have an eyesight problem. If you're on Facebook, look up Nystagmus Network - a really friendly group with loads of knowledge.
Take care.
at the length of that. So sorry!