By default, yes, the employer will deduct 20%, but that doesn't affect your personal allowance, it simply means they don't know how much tax to deduct so they play it safe. Specifically, you're not able to give them a P45 (P46 at best) so it's emergency tax code time. However, you're entitled to claim back any overpaid tax from HMRC after the fact.
If you have an idea how much you'll get annually from each job,.you can ask HMRC to split your allowance between the two. You'll have to nominate an amount for each job, but if both your estimates are less than what you actually earn, you won't pay any tax unnecessarily.