A comment on Wheeloftime.tv and Geeklyinc.com has suggested that Amazon's TV division is the studio that has picked up the rights and is currently developing the project. This is purely hearsay and rumour so should be taken with a pinch of salt, but it does have a lot of logic to it and Amazon had emerged as (potentially alongside AMC) already the most likely home for the Wheel of Time TV series anyway.
"I have a friend who works for Amazon, and he tells me they were heavily involved in negotiations for the rights to Wheel of Time. Jeff Bezos loved the books and felt that wot would be great for their platform."
Based on other information from people involved (if more at the periphery), it sounds like HBO is not involved at all. This makes sense because HBO typically does not pursue multiple series in the same genre simultaneously. Game of Thrones is their current epic fantasy show and they are considering a spin-off as its thematic follow-up. So it is unlikely that they will look at another epic fantasy TV show in the near future anyway.
In addition, the TV rights to Wheel of Time were sold for "a eight-figure sum", so a minimum of $10 million. This is extremely high, much higher (I believe) than the figure that Game of Thrones went for, and there's not too many companies that would be capable or willing of paying that much. This pretty much rules out any of the standard American TV networks or cheaper cable stations, such as ABC, NBC, CBS or the CW. SyFy is also unlikely: although they can shell out quite large sums on show they generally only do that for one show at a time, and that the moment their big tentpole show is The Expanse.
That really only leaves two possibilities: one of the other big cable networks, such as AMC, Starz or Showtime, or a big streaming platform like Amazon or Netflix. Starz and Showtime might be unwilling to spend that much on a show and Netflix are pretty hot on announcing deals as soon as they are done, which makes it less likely it's them. Although Netflix are prepared to spend big money on certain shows (like Sense8 and House of Cards) they are also quite sensible in keeping a lot of their other projects at a fairly low budget, and it's questionable if they would spend that amount of money on just the rights. But they remain an outside possibility.
AMC, buoyed by the recent success of The Walking Dead and Preacher, had strong potential because they would probably quite like an epic fantasy project which can take the fight straight to HBO. They also have a proven track record with Sony TV, which produced Breaking Bad and Better Call Saul for them, and we know for a fact that Sony were talking to Red Eagle about the project a year or so before the rights reverted (or should have done) to the Jordan Estate. This has always strengthened this argument. Fans were also hopeful for this deal because AMC give their shows 16-episode seasons, which is really the bare minimum Wheel of Time would need to get a decent amount of the story done in a reasonable timeframe (say 6-8 seasons). However, there were also concerns that AMC tend to be quite tight with their budgets (The Walking Dead's budget is $3.5 million per episode, which is pretty stingy given its monster ratings; Game of Thrones's seventh season will have a budget close to $14 million per episode, in comparison).
Amazon emerged as the front-runner for several reasons. First, they are looking for a TV drama project to really make people sit up and take notice. Despite critical acclaim, The Man in the High Castle hasn't quite done that for them. The Grand Tour has been successful but they want a drama show that can compete with GoT, Westworld, Stranger Things and The Walking Dead in terms of buzz, watercooler discussion and driving subscriptions. Second, Amazon have been selling Wheel of Time novels worldwide since 1998 and are well aware of the almost-100 million copies the book has sold, the massive pre-orders and buzz around the series whilst it was ongoing. They also have access to detailed demographic information on the people who bought the books and opportunities for cross-marketing. Based on the boom in sales of other franchises, Amazon could make back a chunk of budget just by the explosion in book sales a high-profile TV series would generate.
In addition, Amazon held a creative reorganisation of their TV division in 2016 that took up a bunch of the year and involved, amongst other things, Frank Spotnitz leaving The Man in the High Castle as a producer and a radio silence descending on their new project announcements for several months. This would explain why we initially got news on the TV show and then abruptly nothing since.
Of course, the Wheel of Time TV project may still end up somewhere else or dying in development hell and this rumour remains just that, but it is credible. Hopefully we will know more soon.