MARY POPPINS ANALYSIS
Mary Poppins was actually Bert's nanny long ago and there is a special reason she returns to Bert's life. Let's break it down!
Bert sings: "Because I was afraid to speak/When I was just a lad/My father gave me nose a tweak/And told me I was bad/ But then one day I learned a word/That saved me achin' nose/The biggest word I ever heard/And this is how it goes, oh/ Supercalifragilisticexpialidocious."
Bert explicitly says that, when he was young, he was a scared child who was afraid to speak and had an over-bearing father, until he – as a child – learned that magic word. It's a magical word that really only Mary Poppins knows or uses. There is really no way he could have learned it as a kid unless she taught it to him!
In the movie Bert works as a one-man band, a sidewalk chalk artist, a super-chipper chimney sweep, and a kite salesman! The movie takes care to show how sucky all the jobs are, but no matter what, he manages to keep a smile on his face. It's as if someone taught him how to find the joy in hard work. Perhaps with a spoonful of sugar?! (Yes. The answer is yes.)
In the very beginning, Bert predicts, "Wind's in the East/ Mist coming in/ like something is brewing/ about to begin./Can't put me finger/on what lies in store/ But I feel what's to happen all happened before."
It seems vague, but really he's just being coy. It did all happen before, when Mary Poppins was HIS nanny.
Bert NEVER questions Mary's magic not when she's flying, not when she's turning a cloud of smoke into a magic staircase and not even when she bends the laws of space, time, physics and reality to transport them to a magical chalk-world universe.
It's all old hat to him, because he knows all of Mary's magic tricks from when he was a child! It doesn't seem likely that they would have randomly met on the street and Mary just happened to show her reality-bending, universe-jumping supermagic to some arbitrary stranger.

But why did Bert age and not Mary? She's a magical demi-god who can bend reality to her whim. It feels like a person who can turn clouds of smoke into solid steps to travel around the rooftops of London and can fly with an umbrella might be able to make herself look perpetually young.
Dick Van Dyke plays Mr. Dawes as well as Bert. He's the only actor in the movie to play two characters. Also, he looks just like an old Bert, because he's Bert’s dad!
Remember that lyric from Supercalifragilisticexpialidocious? "Because I was afraid to speak/When I was just a lad/My father gave me nose a tweak/And told me I was bad." It sounds like his father was a real jerk! Know who else is a real jerk? Mr. Dawes, the character who just so happens to be played by the same actor.

There is also a lyric from the song Jolly Holiday that Mary sings which goes, "Though you're just a diamond/ In the rough, Bert/ Underneath your blood is blue!" The term 'blue blood', as defined by the Merriam Webster Dictionary, is "membership in a royal or socially important family."
Bert is a blue blood. Mary knows this because she was his nanny. She helped raise him and she knows his father was Mr. Dawes.
As we know, only people associated in some way with Mary Poppins float when they laugh. Not sure if you know this, but people don't naturally float when they laugh. Before Mr. Dawes finally dies from laughing, he starts to float! In the boardroom! This is proof, within the context of the movie, that Mary Poppins has been in contact with him during his life.
Let's speculate that Uncle Albert is Mr. Dawes' brother. When Dawes had his second son, he named him in honor of his brother Albert. Or Bert, for short. This is also why Mary Poppins knows him as Uncle Albert! He's not her uncle. If he was, why would Bert just be at the house waiting for her when Mary gets there? She calls him Uncle Albert because that's what she's used to calling him when she saw him often during her time as Bert's nanny.
As we've established, only people who have come into contact with Mary Poppins float when they laugh. This has clearly been a recurring problem for Uncle Albert. Bert says, "Last time it took us three days to get him down."
NOW, LET'S TIE IT ALL TOGETHER
So Dawes had Bert when he was already rather old (say 40 or 50, not unheard of). He already had one son (Dawes Jr.) who followed in his footsteps but Bert was shy and reserved -- not the cutthroat personality type needed in the banking world.
Dawes then hires Mary to be Bert's nanny, and Mary tries to work her "saving Mr. Banks" magic on Dawes by teaching him the meaning of family, life, and happiness. But it doesn't work! Dawes is already too old and stuck in his ways.

However, it does work on Uncle Albert, who was a younger, voracious banking world shark. He realized the joy in the world and quit the bank but kept his savings. This explains why he still appears to be well off.
Mary Poppins feels a tremendous amount of guilt for not being able to help Mr. Dawes be a better person and a better father, and for not saving Bert from a destructive home life. This is why she frequently pops back into Bert's life: To see how he's getting along in life, and brings new joy into his world as frequently as she can.