Just some speculation of mine of the franchise's development timeline:
Here is my thought process of the Sonic Boom timeline - and the SEGAbits Sanzaru interview which goes up in a few hours backs up some bits from my speculation:
Project Apollo (before "Sonic Boom: Rise of Lyric"): I'm of the mindset, and the concept art leaked from what we now know as Rise of Lyric supports this, that the project was initially something very different. In 2007 SEGA president Simon Jeffrey stated "We are actually undergoing a fairly considerable refresh of Sonic as an intellectual property, as a character" and "We're not going to be bringing another game out on the other platforms for quite some time, because we feel that it is time to reinvent Sonic, to make Sonic contemporary again. At the same time, even the recent 360 and Playstation 3 games have sold extremely well. Sonic is still an extremely endearing character; there's still a lot of love for Sonic out there in consumerland."
source:
http://www.computerandvideogames.com/161224/time-to-reinvent-sonic-says-sega-boss/Well guess what, in 2008 Sonic Unleashed released. My guess is, SEGA of Japan and Sonic Team disagreed with SEGA West's idea of putting Sonic on pause and refreshing the brand. But it seems SEGA West was persistent and they sides agreed on the side-series idea we have now, a Western developed Sonic Heroes-style game. How do we know this? In 2010, SEGA's Mike Hayes stated "Then you’ve got something we haven’t seen for a while, which is like Sonic Heroes, multiplay-type game."
source:
http://board.sonicstadium.org/topic/4701-cvg-interview-segas-mike-hayes/Sonic Boom: Rise of Lyric: This is when the Rise of Lyric we know now started to take shape. While the team were still pushing radical designs, Sonic Team pushed back (as revealed in BRB interviews) and characters underwent less of a redesign. Also, with the Wii U deal SEGA had, development shifted from multi-platform to Wii U which lead to issues with the CryEngine and cutting co-op in half.
Now here's some more speculation:
Sonic Boom TV: SEGA wanted a Sonic TV series, saw the already in development Project Apollo, and decided to have the French animation OuiDo! studio work in partnership with BRB so that the two could form the Sonic Boom brand. However, since the BRB game had undergone so much development of their own, and the cartoon was finding its own look and feel, there were some elements that didn't line up. the Wii U game had the character designs and both shared NPCs, but the game lacked much of the TV show locations, Sticks, and had more of a snarky sarcastic vibe. Eggman was more of a threat in the game as well.
Basically, SEGA allowed OuiDo! develop what they felt would be the best for their show, and didn't tie them down so much to the BRB game's locations, attitude, enemies, and (Lyric and the ancient robots are absent from the TV show). The two did however attempt some late additions to reference one another - Sticks is an NPC in the Wii U game and the TV show's first episode features Tails' Wii U airplane and the enerbeam. Note that Tails' Wii U airplane is quickly destroyed in the first episode and later we see a totally different plane. Also note that the enerbeams have only made one appearance in the intro of the first episode. I honestly think the first episode's intro was a late addition to the series, bringing in a few game elements.
Shattered Crystal: As revealed in our upcoming interview with Sanzaru, there was very little BRB collaboration and much of the designs and concepts came from the TV show. In fact, the 3DS game has much more in common with the show, from locations and characters to bonus items from episodes (wonder if Buster appears?). The 3DS game likely began development once the TV show had a fair amount of episodes produced, allowing the team to almost exclusively draw inspiration from the show. The only BRB connections are Lyric and the enerbeam.
Summing it up: The creation of the Boom franchise likely went like this: 2007 SEGA West wanted to drastically reinvent Sonic - remember, Simon Jeffrey was the SEGA president obsessed with reinventing classic SEGA IPs (Golden Axe Beast Rider, Alien Syndrome). SEGA of Japan and Sonic Team said "no", eventually Simon left but the idea lived on. In 2009 SEGA east and west agreed that SEGA West could develop a side-series with Sonic Heroes style gameplay. Meanwhile, Sonic Team was releasing Colors, Generations, and SLW. SEGA's Nintendo deal leads to the spinoff becoming a Wii U exclusive, which threw a wrench in development. Around this time, SEGA also wanted to make a TV series and either the series was initially developed separate and influenced the Wii U game changing direction, or SEGA felt the TV series had a better idea of reinventing Sonic than BRB and had both work together. This leads to inconsistencies between the show and the Wii U game. The 3DS game starts development much later, and almost exclusively uses the TV show as inspiration, making it much more in line with the show and comics.