Brexit is chasing all the "rats" round the yard, chasing them out, blocking all the ways in....except leaving one wide, wide open (the Irish Border). So there will be zero saving in chasing down illegal immigrants - except that you are actually adding to the numbers that need chasing on top of the r/o/worlds already.
If someone was - after Brexit - as desperate to come here as they are now, then that Irish route will be wide, wide open, as long as they are EU citizens and legally entitled to be in the Republic.
Even if the job of finding and reporting them is given to the employers and landlords, there will still be substantial cost in getting them, examining the case and then if they appeal the courts, then the appeals, and finally getting rid.
And next month back they come via Ireland....