EU procurement rules allow discretionary exclusion of companies "where they have committed offences or undertaken activities relating to misrepresentation, undue influence on procurement procedures, grave professional misconduct, agreements to distort competition or demonstrated significant or persistent deficiencies in the performance of a public contract". That seems like ample grounds for the UK government to exclude their usual outsourcers; that they choose not to do so is purely the fault of the UK.
As for your suggestion that UK voters will be able to affect this stuff more after Brexit, I have seldom seen such stupidity. In case you haven't noticed, cabinet ministers - who are theoretically responsible for this stuff - tend to get given safe seats where a donkey could get elected if it was standing on behalf of the correct party. That's not going to change.