I have done more tests today, using different HDMI cables, and different apps.
The problem is actually quite complicated.
If the Ipad is switched on with the HDMI adapter already connected, that error message appears, and there is never any video output. Never. No way to get it to work, short of unplugging the HDMI adapter and power cycling the Ipad.
If the Ipad is switched on and the HDMI adapter is plugged in sometime after that, and one goes straight to Iplayer, and does nothing else, then it works. I do notice a 1 or 2 bars drop in the wifi signal level though, compared to no HDMI adapter+cable connected, so there appears to be a reduction in the wifi signal due to the HDMI activity, even when no video is actually being played through the HDMI cable. So this looks like RF emissions from the HDMI output. I wonder if Apple ever did RF emissions tests on that HDMI adapter... (Conditions: Draytek 800 access point, latest 1.0.3 firmware, 5m away, 2m HDMI cable, Sony 1080P TV).
If one does the above and then tries other apps, it gets more complicated. Normal screen mirroring continues to work fine so you can view photos etc on the TV. But if you try video playback from non-Iplayer apps, e.g. Vimeo 720p (running in a browser), while the Iplayer app is in the background, the Ipad gets well confused. The wifi basically stops working, and never gets working again even if you close down all the other apps. The only way I found out of this is a power cycle of the Ipad *and* a power cycle of the wifi access point.
I had similar issues with other wifi access points (Draytek and Linksys) but back then I did not spend much time on it.
I think that, apart from anything else, the Ipad gets confused when one tries *video* playback from one app while another *video* app is in the background. That seems to screw up both the HDMI output (which on one occassion showed both a piece of the Ipad screen, and the video in the background) and the WIFI subsystem.