Not quite. Bethesda hasn't just started working on ESV when Zenimax aquired id. Bethesda is also very familiar with Gamebryo, they aren't just going to drop it. It's likely they'll get advice and some bits from id's code into ESV, but they aren't going to switch engines entirely. Ashley Cheng also stated that they are comfortable with Gamebryo right now, and are not planning to switch.
Yeah, there was a team working on Fallout 3 while the rest of Bethesda finished Oblivion, and shortly after Oblivion was released Fallout 3 was able to enter full production. Chances are Bethesda accomplished this with their next title as well, getting through most of the pre-production while Fallout 3 was being finished - which would mean it's been in probably been in full production for nine or ten months. This would mean that the groundwork for the engine was in place before id joined Zenimax. Switching to idtech would scrap a lot of work they'd already completed which doesn't make much sense.
Don't hate me for this, but i agree with GHE, it's not yet time for them to change their engine, maybe TES VI but not TES V, not only have they started getting seriously good with it, but they probably don't have the time for the new engines, they probably started work on TES V from the end of TES V, from there they probably had a small team working out the basics of TES V like story, background ideas etc. Not to mention the considerable amount of time they would need to get good with the other engines, it would start taxing heavily on development progress.
Fallout 3 is using an upgraded version of the Oblivion engine, and Fallout New Vegas is uses a tweaked version of the Fallout 3 engine. However I think FONV is the last time we'll be seeing it. I suspect Bethesda's next game will use a new engine and be released for next generation consoles (and the PC). It's certainly possible that they'll try and get one more game out of the engine, and it wouldn't surprise me to see this, but I just think a new engine is more likely.
Assuming they've started work on a new engine already it's almost certainly Gamebyro. They had no close relationship with id software but they would have started this and, as I said before, it doesn't make sense to scrap what they've accomplished. However, the next time they start an engine from scratch I think there's a good chance that they'll use an idtech engine. While they do have a lot of familiarity with Gamebyro they can get idtech for free, so it is not only more economical but it would also serve as 'advertisemant' of id's engine.