Is he? The movie is set in 1910, and conscription in Britain during WWI applied to men aged 18-41 and was in effect through 1918; If Michael Banks was eight (the age of the actor at the time of filming), he would have either been recruited at the very end of the war or narrowly missed it all together.
Boys under the conscription age frequently attempted to join the war. Many succeeded. Worse still, when generals started to realise the dire situation for Britain, they started illegally allowing it.
Even in real life the actor who played him died at 21 (of hepatitis).
Even had Michael not died in the war as a soldier, he, Jane, or any British child of that generation (as nonfictional children had) would have been affected by the war(s) to come one way or another. Particularly where the bombings of World War II were concerned.
Bert is definitely old enough to have fought in WWI.
What happened to all those people and talking animals in the chalk-drawing world when the rain came?
Maybe the painting was just the gate to that world?
The use of children for chimney sweeping was outlawed decades before 1910, in fact the equipment Bert is carrying when he meets the Banks kids after the bank run was created specifically to replace child labour.
No matter how greedy, you'd think a veteran financier like Dawes would have the common sense to let go of a measly tuppence immediately, rather than let some kid's hissy-fit provoke his bank's near-collapse.
It shows how out of touch the bankers are, going on and on about imperialistic accomplishments ("Railways through Africa, dams across the Nile, fleets of ocean greyhounds, majestic self-advertising canals, plantations of ripening tea."), things that mean absolutely nothing to an eight-year-old boy. If George or the other bankers tried to explain to Michael that by investing, he'd get more money for things he'd like to do, such as feeding the birds, maybe Michael would've been more open about it.
Well, he is a pretty doddery old man. Maybe that's why he's not quick enough on the uptake to realise why it would be better not to cling onto that tuppence. Is it any more plausible that a riot in a bank could happen as easily as this? We're also talking about a film with flying nannies, people jumping into pavement drawings and a retired Admiral whose neighbours tolerate him firing a cannon off his rooftop several times a day.