what are the legal and regulatory implications of a dental hygienist
carrying out the procedure of tooth bleaching and how could this have
implications for the registered dentist under whose direction this was
performed?

Questions that ask about legal and regulatory matters are seldom
answerable without knowing the location involved. Laws vary from place
to place.

In addition to giving us a jurisdiction to work with, I suggest that
you increase the price. It is not likely that a question like this one
can be well answered in the five or ten minutes that most Researchers
would devote to a $2 question.