I had been following this thread just for my own education, but today, I was motivated to look at my own photos of the three species. I have only a couple of encounters with Cinnamon-breasted and Blue-chested and, like everyone, lots more with Little.
After looking at my photos of Cinnamon-breasted from near Mount Kenya (at a distance) and western Uganda (much closer), I was going to agree with Cinnamon-breasted Bee-eater.
But then I looked at my photos of Blue-chested from Ethiopia (two locations).
And...hmmm.
Valéry's argument that it isn't Blue-chested, but is Little - now amended to 'maybe it's Cinnamon-breasted after all' - is actually based mainly on location, not on the actual photo.
None of the books (the three main ones I have anyway) shows Blue-chested in the location that the OP says he saw it (about 100km north-west of Nairobi). However, it is shown in two disjunct populations in Ethiopia, and from Lake Victoria in Kenya across to the west African coast. The OP's sighting would be roughly in the (actually quite small) 'empty space' between these two populations.
As Valéry himself says, the (substantial) white cheek patches don't fit with Little but fit with the other two possibilities, while the fairly obvious area of blue above the eye fit with Little and Blue-chested, but not very well with Cinnamon.
Adding these two features together, Blue-chested should be the preferred candidate, but seems to be ruled out of consideration just on location.
I think Little can be ruled out on the white cheeks which are much larger and more obvious on this bird than on even a slightly unusual Little.
For Cinnamon-breasted, Stevenson and Fanshawe (and Sinclair, same picture) shows a uniform cinnamon colouration from the bib to the vent, but in my experience of these birds, there is a strong orange area below the bib, and then the colouration gets paler as you go down. The vent is actually a very washed-out yellow. Zimmermann's illustration shows more of a gradient of dark to light as you move from the head to the vent (but it's still too dark overall, I think). But the bib is clearly black.
But the OP's photo shows a blue-ish tint to the bib itself, as well as the blue above the eye. I briefly wondered if this was because the OP used flash to take the photo - so the blues were a photo artefact.
But when I compared the OP's photo with a Blue-chested from Ethiopia, they look almost identical to me. And I don't think anyone is suggesting that Cinnamon-breasted strays to Ethiopia?
My Langano bird has significantly more blue above the eye than the OP's bird, but the OP's bird seems to have a wider spread of blue than Cinnamon, and also splotches on the bright head.
As regards habitat ruling against Blue-chested, we all know that the weather (rainfall patterns and so on) have been peculiar in this part of the world recently. So the Blue-chested in this location might be possible. After all, if it was there, and people saw it, then they would probably simply put it down as Cinnamon-breasted or maybe Little - because 'Blue-chested doesn't occur here'.
Would anyone like to reconsider it?
Photos:
1. Cinnamon-breasted, Uganda, to show the underpart colouration gradient.
2. Blue-chested from Langano, Ethiopia
3. A composite of the OP's photo (left) and my Blue-chested from Langano (right) They look a good match to me.