Thinking about synergy, one of the big things people have to look at is where do things fit into the overall scope. As much as we talk about Hulu and Sky, there still is the fact in how the film/tv shop IPs will be leveraged. Upon further analysis, I have observed how NBCU and Disney may leverage some of these IPs as well as why.
Fox's history has been built upon producing award winning dramas in film and television, cult films and tv shows, and the largest lineup of movie musicals of all the studios. Prior to the buyout by Newscorp, they produced shows and distributed films for ABC (owned by Disney), CBS, and on occasion NBC. Some of their most successful properties include Die Hard, Black Mirror (under Edenmol Shine) Home Alone, Miracle on 34th Street, Buffy the Vampire Slayer, Firefly, Modern Family, The Simpsons, and their collection of Rodger and Hammerstein films they own the rights to.
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NBCU: Overall for content leverage, the biggest win content wise for NBCU in regards to the potential Fox buyout will be obtaining rights to things they can leverage for the Dark Universe such as the League of Extraordinary Gentlemen, X-files, Black Mirror, Rocky Horror Picture Show, and Buffy the Vampire Slayer as well as filling gaps in alternative programming that Focus Features/Universal Pictures has not been able to penetrate (sci-fi and adventure films).
Currently, NBCU is trying to increase animation output and currently lacks all adult animation (Archer, Simpsons, American Dad, Futurama, Brickleberry, Bob's Burgers). Sony, Fox, and WB currently have access to this gap with shows on comedy central, cartoon network, adult swim and other network channels with massive success. Additionally Fox winning quite a few bids on upcoming children book franchises (they have 14 animated features based on books and children IPs coming out here soon) will be a game changer for any company who obtains them. Though many people believe Universal will divest of Blue Sky and Fox Animation, I believe they will keep both (restructure so Blue Sky is a subsidiary of Fox Animation) and have them work independently as they are the producers of those adult shows and to maintain rights to continue to develop those animated franchises they recently gained access to.
Focus and Fox Searchlight will both remain as indie/arthouse studios with a focus to create unique stories. Focus will focus on stories developed and told in the UK as they currently operate and Fox Searchlight will remain focused on the US with similiar stipulations but essentially doubling the content output.
In terms of television, 20 Century Television and Endemol Shine likely will be absorbed into Universal Television, Universal Cable Productions, and Universal Alternative Studios. There is already so much overlap that it doesn't make sense to keep these independent outfits. Endemol Shine will be strongly leveraged by Universal Alternative Television Studios and UCP for their strength in creating popular reality shows as well as their independent production companies who have produced shows like Black Mirror, Big Brother, and Wipeout. Endemol also has an active gaming division focusing on strictly mobile games development which can be leveraged by Universal Branded Entertainment. Universal Television will probably maintain or try to continue push out the level of comedies Fox puts out to put onto Hulu as well as utilize their Star IPs to potential open Hulu in new regions as its only currently available in the United States and Japan (Of note Japan Hulu literally has everything from Game of Thrones all the way to every episode of Family Matters). The reality shows will be used for the Hayu app which is currently doing better than expected in other countries.
Lastly, in terms of 20th Century Fox Films being independent or merging with Universal Pictures, this is where things get harder. Mainly due to certain rights reverting back to another company if certain things happen. If 20 Century Fox is no longer a parent company or independently operated, essentially all of the X-men/Marvel rights currently being produced and distributed by Fox go back to Marvel. Universal still has some production/distribution rights for Namor (who is tied to Deadpool if you read the comics) and sole distribution Hulk rights. If Xmen, F4, and related IPs all returned to Marvel, that would drastically reduce the value of Fox in terms of distribution. However, with them operating seperately and independently, it causes complication with the release dates and cycle as you don't want one blockbuster from one studio you own to cannibalize another blockbuster. To mitigate that, we may see a change in genre focus for either studio with Universal Studios maybe going back to their roots and focusing on pushing Horror, action, and comedic films while Fox may be used mainly for musicals, sci-fi, and dramatic films.
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Disney: In terms of content the biggest win for Disney will likely be all the adult fare that they currently lack which they gain by Fox ownership. Fox Searchlight and the shows produced by FX will be the ones that are leveraged the most to give them a more robust streaming service than what they currently have. In addition the film rights to the musicals, such as 9 to 5, Working Girl, Sound of Music, and the Rodger and Hammerstein library is something Disney has wanted to leverage in the past for their theatrical division.
WDAS and Pixar have a level of respect that unfortunately will be hurt by owning Blue Sky and Fox Animation. At the same time, Disney require those same assets as Universal mainly as it provides them that adult content but Fox Animation currently lacks any level of respect within the animation community nor has ever been recognized as quality which hurts.
In terms of television, this will be the real winner for Disney. With their already built in market for pushing out franchises, Disney can easily reveal cult favorites such as Firefly and Buffy with maximum success massively increasing their valuable IP franchise numbers. They also would very much utilize Fox produced comedies they already distribute for syndication and oversea sales which they don't currently get a dime of. It will also bolster their reality tv offerings so they will no longer require to air the problematic bachelor tv shows that clash harshly with their Disney image.
Lastly, I'm going to assume unlike Universal, Disney will fully absorb Fox and have it act as Touchstone. They would not face the same problems as Universal as they already own Marvel and it would work to their benefit not to cannabilize their own films. As stated they don't carry adult fare and this gives them a major studio to do just that.
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Disney synergy is a lot easier to see as it lacks a lot of content that Fox provides, however NBCU has the infrastructure that is enhanced by the content Fox creates. One feels the gap while the other gets made better.