In reply to Chris Craggs: Actually, nightjar did briefly cross my mind: the general shape was right ie falcon/cuckoo-ish, and the colour could work, but just about everything else is wrong. It was the middle of the day, and in Edinburgh which is (a) outside their range and (b) a city, albeit Blackford Hill and the Hermitage have quite a rural feel to them. It just seems far too unlikely to have been a nightjar in that location, or at that time of day.
I know what a jay and a collared dove look like and it wasn't either of those. It was too sleek for a dove and didn't have a collar. The flight was much more fast and direct than a typical corvid - and jays' tails are black, with a distinctive white rump which this bird didn't have. Note that I saw it from above, not from underneath.