After returning from two weeks' holiday where I'd used only Wi-Fi and not 3G at all, the latter just would not connect.
The resolution is to set up a new APN:
How to configure Internet Settings on your HTC Desire HD.
Press the Menu key
Tap on Settings
Tap on Wireless & Networks to select.
Scroll down and tap on Mobile networks
Tap on Access Point Names to select.
Press the Menu key
Tap on New APN
Tap Name then enter T-Mobile Internet select OK
Tap APN then enter general.t-mobile.uk select OK
Tap Username then enter t-mobile select OK
Tap Password then enter tm select OK
Tap APN type then enter default select OK
Press the Menu key
Tap Save
Press the Home Key
Your phone is now ready for use