Swifts tend to fly higher than swallow and house martins, but the biggest giveaway is the noise they make. From almost the same page you linked you can listen to the audio file. A swift call is a screech, even a scream. You can quite often hear them before you see them (they tend to occur in decent numbers when the noise is almost non-stop) Swallows and house martins both make a softer, more twittering sound.
Visually the swift has much longer (and more curved) wings. It also has a longer tail (excluding the swallow's very thin streamer feathers which often can't been seen with the naked eye in high flight). The fork appears deeper too. To the naked eye, a swift is uniformly dark, while house martins and swallows both show significant pale areas on the underside (and may even come low enough for you to see this). The flight is different too -- swifts arc through the sky rather than flapping as much as the others.
The RSPB page sums up the difference in the nests. Basically if you see a mud nest it's not a swift.
Update: the British Trust for Ornithology has posted a video on YouTube addressing just this question.