Flights to and from the UK (and the USA) require Advance Passenger Information, which is submitted to the government by the airline. This information includes the passenger's name as stated in their passport.

When Ms Higginbotham checks in for her flight, the name as stated in her passport will be captured by the airline and associated with the ticket, for submission to the government.

UKBF agents probably can find full details of the air tickets if they want to, but I believe that they don't usually do so; they simply scan your passport and (for US citizens) will ask a few questions and that is all.

The only party that might care about the name is the airline; and therefore, I suggest that you don't use Maggie but rather Margaret since that is her passport name. The middle names can be omitted.

Her US driver's license is of no relevance unless she intends to rent a car in the UK. Since her DL name is Maggie, I would recommend that any car hire is booked under that name. However it is extremely expensive if not impossible for anyone under 25 to rent a car in the UK from a main agency.