The Australians could do it and solved the arrival of illegals very effectively, no reason the UK can't.
IIRC the Australian solution was to establish a deal with one of the countries off its north east coast. Any illegals were picked up and transported there. Word was also spread at source that even if you got to Aus you would not stay.
As I said, I would place our problem is letting them loose in the population. It means they're pretty much sorted from that point on as they can just abscond and the chances of finding them again are very very low.
I suspect that's the cheapest option though. As any other option is bloody expensive. Eg: detaining them until their identity can be confirmed (or they supply the details themselves). The sheer number of detention facilities and staff needed to ensure that will not become some hell hole are massive, and that's going to cost. To give you an idea last year's arrivals alone would have meant a 10% rise in prison population.
There in lies the problem. We're a bit broke at the moment. The chances of the politicians being negatively effected by illegal immigration is damn near zero. So there's no incentive to fix it.