Prince Philip's comments about the Royal Family's finances on "Meet the Press" are depicted as causing political blowback that led him to propose making the BBC documentary. In fact, he did not appear on "Meet the Press" until November 1969, after both the release of the documentary and the arrival of his mother in Buckingham Palace.
At 48:57, in the newsroom, an extra picks up an old fashioned corded telephone handset. At first glance, it may appear that the actor has never handled this kind of handset before, but if you watch closely, he maneuvers the handset in a certain way to avoid getting the cord tangled up or caught on anything on his desk. That is the reason it looks so clumsy.
When Princess Alice takes the gemstone to a dealer for appraisal, he puts on some strange optical device which seems to have been fabricated from a pair of opticians' trial-case lenses of deep concave power. Presumably it is supposed to represent a magnifier but would in fact have the opposite effect, making the object look very much smaller and further away.