Alex Doyle has never attempted to grow a mustache. Actually, there's no evidence to support that. We can say that Alex has never successfully grown a mustache. She may have attempted and failed. It's impossible to prove otherwise. None of that has anything to do with her involvement in the TARDIS in Orbit project, however.
She grew up hearing stories of The Doctor from her father, and one day, while flipping through channels found herself watching BBC America. A moment later, she was staring at a rather odd looking upside-down dust bin with a plunger sticking out of it, and she actually thought "My father told me of these creatures!" She was suddenly filled with terror.
She immediately went to Netflix and watched 5 seasons of Doctor Who in a week. This of course left her brain somewhat subject to suggestion, and probably explains why she readily agreed to help put a TARDIS in orbit around the Earth.