[noggin]

Here are my observations with a Fire TV Stick 4K connected to a Sony 49XF9005 via a Denon AVR X2400H and an HD Fury Vertex (which tells me a lot about the HDMI signals being fed to the Denon)

Amazon Fire TV Stick 4K is running Software Version Fire OS 6.2.5.8.

Display / Audio settings are :
Display
Video Resolution : Auto (up to 4K)
Match Original Frame Rate : On
Colour Depth : Up to 12 bits
Colour Format : YCbCr
Calibrate Display : No Scale
Dynamic Range Settings : Adaptive

Audio 
Surround Sound : Best Available 

The default gui output is : 2160p 59.94 BT709 12 bit 4:2:2 YCbCr (593MHz HDMI clock with HDCP 2.2) and with DD+ Audio

iPlayer
UHD stuff, like Dynasties, which should be output at 2160p25 or 2160p50 Rec 2020 HLG is output as 2160p59.94 Rec 709 SDR 4:2:2 YCbCr.  The colours look drab and washed out. Forcing my TV into Rec 2020 and HLG mode it looks much better - but has the classic '25p at 60p' motion artefacts.
My guess is that the iPlayer app has yet to be rewritten to trigger the frame rate changing (you don't get the 'Adjusting Frame Rate...' message you get in Prime Video), and also I suspect that there isn't a flag to switch the HDMI output flag to Rec 2020 being sent to the OS either?
(Interesting that the iPlayer menu also looks horrible when you overlay it on output that is supposed to be UHD Rec 2020 output - so the app itself appears to be switching internally to Rec 2020 for the GUI)
(Conclusion - no frame rate switching or gamut and EOTF switching implemented)

Netflix
Lost In Space is shot 2160p23.976 and available in at least HDR10 or DV on Netflix.
It's output as 2160p59.94 Dolby Vision in Rec 2020 YCbCr 422 12 bit.  So the colour gamut and output EOTF is correctly set, but the frame rate isn't adjusted.
1080p25 HD SDR Netflix content (like most European TV drama) is also output at 59.94Hz in Netflix.  The SDR EOTF and Rec 709 gamut is correctly signalled.
(Conclusion - no frame rate switching, but gamut and EOTF switching implemented)

Amazon Prime Video
Jack Ryan is 2160p23.976 and available in at least HDR or DV on Netflix.
It's output at 2160p23.976 12bit 4:2:2 Rec 2020 Dolby Vision - after an adjusting frame rate message.  This is the only content that is played correctly so far.

If I could find Grand Tour in UHD on my Amazon Fire TV Stick 4K I'd report whether it was output at 2160p25 or p50 with HDR.  However I can only find the HD version - which is output at 2160p25 BT 709 4:2:2 12-bit SDR...
(Conclusion : Frame rate, gamut and EOTF switching enabled)