I'm still doubtful that it's Parliament that must remove the titles. In researching, it sounds like styles & titles are entirely the queen's prerogative. In fact, it sounds like her wishes need not come in a formal document & can be made known by press release or even verbally:

The queen has not referred to Harry, MM, nor Archie with by formal style or title since before Megxit.

After Megxit, Harry did say to call him 'Just Harry.'

I'm not sure we know the whole story and I'm not sure that's even important to the facts of the matter of their titles.

Source: https://www.heraldica.org/topics/britain/prince_highness.htm#Wessex

This is a very good resource on the rules and history of the royal styles and titles.