and as discussed only the other week, there is not really such a thing as "illegal immigration" only "irregular immigration".
There is nothing illegal about travelling to a country and seeking asylum.
I know what you're meaning, JP_, but there is illegal immigration.
However there is no illegal asylum seeking. It is not illegal, and everyone has a right to arrive in another country and claim asylum, for whatever reason, and irrespective of how you arrived.
There is some illegal immigration.
As Notch7 says, visa overstayers are illegal, but there's a small fraction of them, and they are probably mostly students, well integrated into society, having been educated in UK. Although many cannot wait to return home to put their newly acquired knowledge, wisdom and awareness to promote improvement in their home country.
But illegal immigration is the small number of immigrants who sneak, or are smuggled into the country, perhaps in the back of lorries, etc. and make every attempt to evade detection. Again, that will be a very small number of people, but of course the very nature of their arrival and avoiding detection, means no-one can put a number to it.
There also may be failed asylum seekers who abscond detention. Perhaps the numbers exist in government websites.
There are also the failed asylum seekers, who are released from detention, but no offer of repatriation is offered. So they have no alternative to remain in UK, even though they are failed asylum seekers. So officially they would be classed as illegal immigrants.
Then there are the Windrush generation, who strictly speaking might be classed as illegal immigrants, but to repatriate them is inhumane.