The thing to understand is that Bobbie was in a bit of a reckless mood anyway as she stormed through Kew Gardens. She'd had to leave Lady Wickham and the theatrical agent in a slightly awkward moment of pleasantries-giving-way-to-opening-negotiations in order to find someone to read the part of Hiram O'Toole, the ancient ostler and Greek chorus-chappie. Bucky Buckingtithe was supposed to do it, but Bucky was currently sulking unhelpfully over a tonic water, and she needed a subsitute in short order if her mother's play stood a chance of selling.
She was therefore in no mood to brook resistance from the floppy-haired bird she came across, futzing with some sort of machinery in the Color Spectrum. He insisted he was in the middle of something frightfully important, as men always did, that he couldn't spare the time, and that furthermore he wasn't an actor but some sort of medico, which seemed exceedingly unlikely to Bobbie. She pointed out that if he didn't want to be taken for an actor he shouldn't wander around being goodlooking in erstatz cricketing duds. This thought seemed to strike him, and with the assistance of a nearby ginger schoolboy she managed to drag him over to their table at the Orangery.
Bobbie explained along the way that the part was really quite easy, but Bucky, who was her current fiance, had kicked up rough over some silly little observation she'd made about his tie, and she was as a consequence now giving serious thought to returning him to store. (The schoolboy: "Can you do that?") The reading would only take a few minutes, and then the Doctor could get back to contacting his friends; she'd be terribly grateful and might even let him give her lunch tomorrow if he liked. He protested, but followed her like a lamb.
At their table, her mother looked immensely relieved as she introduced the Doctor, over a certain amount of spluttering, as her old friend Freddie Fortescue, who was in "Hello, Girls!" at the Splendide. The schoolboy bunged out his own name at this point as well, so she was forced to claim him as her Hungarian-Irish cousin off on half-holiday, while shoving a script into the Doctor's hands and thanking him sweetly for adding a touch of professionalism.
For a wonder, he wasn't at all bad at reading lines, although he did have an annoying way of staring at the page as though he didn't quite believe what he was seeing or that he was there at all, and twice she had to glare at him when he took out a pair of glasses or seemed about to break out into editorial commentary. They were just at the part where Hiram reveals that the impoverished hero is really the son of the cruel Lord Strathmorelick when one of the nearby potted oranges exploded.
The next five minutes were rather full of incident. Bobbie was attempting to ward off with a few well-aimed eclairs the creatures with the tinny voices who looked rather like marionettes made with artificial flowers and pipecleaners, and who were very fortunately, it seemed, deathly allergic to custard cream, when the Doctor managed to do...well, something...and all the alien pipecleaners disappeared at once. In relief, and with a certain awareness that Bucky was just watching with a gaping mouth and chocolate icing all over his tie, Bobbie sprang at the Doctor and kissed him.
It was one of the most amusing afternoons she'd had in some time.
(The Doctor did end up inviting her to lunch the next day, as it happened, but since the first place he took her in his police box thingamagig looked like some sort of quarry, she decided his taste in restaurants was an insuperable difficulty to any romance and made him drop her off at the Ritz.)