I heard the guy on last nights program say that you can get an emergency travel document within 24 hours from the Nigerian embassy so I wouldn't be surprised if most embassies do the same. If the illegal imigrants knew the system had changed and they couldn't just work the system by claiming they hadn't any documents and they were likely to spend quite a while in a detention centre with the same end result (deportation), then I'm sure they'd would be queuing up to show their passports or apply for emergency travel documents once they had been caught.
I think there might be some initial extra expense until the problem was brought under control, but in the long run I think it would save millions of pounds and improve the job prospects of those who were born here or are legally entitled to be here.
Iain.