The data is taken from the section where Pippin looks into the Palantir. At this point one Nazgul is already on route to Isengard, and a second is sent at the point Pippin finishes looking in the Palantir. These two Nazgul are also observed/felt by Frodo, Sam and Smeagol as they are hiding near the Black Gate.
It is helpful having these two events, as it enables comparison between the speeds of the two Nazgul (riding on Winged Beasts), helping to sort out a few ambiguities as to the time when Pippin looks into the Palantir.
The main assumption is that it is about 11 pm when Pipping looks in the Palantir. At this point the second Nazgul is sent to Isengard, with the first one already on the way. Assuming it takes about 1 hour for Gandalf to interrogate Pippin and the subsequent debate, the first Nazgul then flies overhead at 12 midnight. Sam/Frodo/Smeagol notice the first one pass over at dusk.
A few words on distances: Sam/Frodo/Smeagol are at a point near the Black gate which is approximately 180 miles from Baradur (where the Nazgul are if they are sent by Sauron, presumably), and Pippin is about 600 miles from Baradur.
Assuming dusk is about 6pm, the first Nazgul takes about 6 hours to travel from Sam/Frodo/Smeagol's position to Pippin's. The distance being about 420 miles. This gives a speed of 70 mph.
Sam/Frodo/Smeagol notice the second at "one hour after midnight", so assuming it is released when Pippin looks in the Palantir, at 11pm, it takes 2 hours to travel about 180 miles. This gives a speed of 90 mph.
So after all that, we have two speeds: 70 and 90 mph. It can be expected that the second would be slightly faster than the first, as it was 'rushing with terrible speed', and it was on a more urgent erand.
So, to summarise: The speed of a Winged Beast is between approximately 70 and 90 miles/hour. This speed is ladened with whatever weight a Nazgul is (anyone want to estimate!), but presumably only the clothing needs to be taken into acount as they are not really 'there'.