OK then - tested with Three.
The router gives some handy diagnostic information.
Without antenna
1 PLMN: 23420
2 Service status: Valid service
3 RSCP (dBm): -92
4 ECIO (dB): -5
5 RSSI (dBm): -91
6 Roaming: No
With antenna
1 PLMN: 23420
2 Service status: Valid service
3 RSCP (dBm): -75
4 ECIO (dB): -6
5 RSSI (dBm): -77
6 Roaming: No
I only ran a few tests with each, which were all fairly consistent and the above is representative and fits with "normal" performance for Three round here.
The antenna does improve the signal strength - the lower the dBm number, the better, 92 is borderline poor and 75 is fair to good (from memory). The most obvious effects are seen in the latency and the upstream.
What is slightly unfair as a test is that "without antenna" has the router sitting on the window sill, whereas with the dongle, I'd put it on the top of the window outside facing the cell.
I do believe the antenna is pointing at the transmitter fairly accurately, as the guy could see it with the naked eye from the roof, and fine tuned it with binoculars. Both tests see the router connecting to the same cell e.g. even without the antenna, the device latches to the nearest one.
The conclusion I reach is that while the antenna does help the performance, the difference was possibly worth the circa £300 I spent on it + the fitting it to the chimney given that it was our primary broadband connection. It may also make a difference in poor weather. And it means you don't have to "bring the dongle in when it rains", it's a permanent, tidy setup. Though from these brief tests, it isn't much different in performance terms to having the dongle in its sweet spot. But then you can't do that when it rains.
For anyone reading this wondering why they can't get theirs to work - you have to plug the antenna for 3G into port #2 which is the left hand one on the back of the unit (for 4G you'd plug in a pair of antennae).