Possibly. On one of the forms (I think it's Law Society student membership), referees need to disclose whether a student has been found to have plagiarised. This does not automatically stop the student from pursuing a legal career, but it will make it necessary for that student to make a special case (individual application) to the Law Society for admission.
Similarly, the Bar council requires certificates of good character, and having been found guilty of plagiarism (or, indeed, any other form of unfair means) could mean that referees are unable/unwilling to sign off the form, at least without disclosing the plagiarism.
But, as far as the competition for places is concerned, having plagiarised won't help. Some people do get away with it - but they are only storing up difficulties for later on in their careers when they realise that they can't actually do what people expected of them, having cheated their way through an exam.