You have a couple of good options. First of all, if you want both the Twitter website and app are both quite accessible, the website has a lot of hotkeys that let you go through tweets and interact with them. That being said, I wouldn't recommend using the official iOS app or website, not because they have accessibility issues, but they push a lot of promoted content at you, so you'll get things like advertisement tweets and posts from people Twitter thinks are interesting that you might not necessarily be interested in following. With third party clients this doesn't happen, and you'll only see people that you follow.
As far as third party clients go, like Draq said the main ones people use on Windows are chicken nugget ($15), and TWBlue (free). They're both quite similar and revolve around what people are calling the "Windowless experience" where they have a ton of hotkeys that you can use in any application to read tweets without leaving it. For example, holding down windows and alt, then pressing up and down arrows moves you through tweets, no matter where you are. They both allow you to perform alll Twitter functions, reading and writing tweets and direct messages, following and unfollowing people, performing searches, using lists, seeing what's trending, etc. Additionally, they have some features that are useful specifically for the blind. They both can play any audio files linked to in a tweet without leaving the client (Nugget also supports playing Youtube videos), and they let you attach audio to a tweet yourself. They can also do OCR on an image, translate tweets, and use a third party service called Twishort to write tweets longer than 140 characters with the ability to automatically view the full tweet inside the client (though I personally hate this feature as if you're on anything other than TWBlue or nugget you have to click a link to see the whole tweet which usually only comes down to a few more words.)
One more option you have on Windows is OpenTween, which is the one I use myself. This is a client not made for the blind, so to use it you have to switch to its Window. However, it still has a lot of hotkeys and can also do everything you'd need to on Twitter (minus the built-in audio player and Twishort support). The main reason why I use it is stability and resource usage. Both TWBlue and Nugget use a lot of RAM and sometimes seem to have issues getting tweets (at least for me). Nugget in particular has this feature designed to keep every received tweet in a database, which can get big fast. You can turn this off, but last time I checked this caused other problems. All that being said there are a lot of happy users of both accessible clients, and Chicken Nugget has a 30-day trial, so you can just try all of them out and see which one you like the most.
FOr iOS, you also have plenty of options. The one I would recommend is Twitterrific, which is free with an IAP to unlock certain features. It has really good VoiceOver support and makes really good use of the "actions" option on the rotor to quickly let you reply or retweet something, but also click on links, hashtags or usernames in a tweet. Another good iOS option is Tweetings, which also has really good VoiceOver support. While it doesn't make as good use for actions, it has a massive amount of options to configure, such as completely disabling images, changing how the timeline is sorted, or automatically splitting up tweets longer than 140 characters. Again, I would just suggest you give both a try and see which one you like more.
The twins of Mammon quarrelled. Their warring plunged the world into a new darkness, and the beast abhorred the darkness. So it began to move swiftly, and grew more powerful, and went forth and multiplied. And the beasts brought fire and light to the darkness. - from The Book of Mozilla, 15:1