That's incorrect. The documentary was biased to an extent and I don't think its fair to make sweeping statements regarding immigration. Whilst immigration does have its benefits, it doesn't mean the country is completely reliant on immigrants.
We've grown to rely on immigrants to work in certain sectors and yes, if they left all of a sudden that could cause chaos on a certain level. Over time however, society would re-adjust itself.
The majority of the immigrants do the labour-intensive jobs many people aren't prepared to do. They cannot however, carry out other key services (those not educated or qualified under HSMP etc) that require professional qualifications e.g bankers, legal services, architects etc. Its the advanced level of these latter services that enable England to be a developed country as opposed to a developing one that is, as you say, a desert in Africa.