There are plenty of intersections in Toronto where you have to wait at least a minute if you press the "beg" button.
I've never seen an implementation of the system locally where if the traffic light for cars is green but the pedestrian signal is "don't walk" (because no one pressed the button before the drivers got the green), it will automatically turn on the "walk" signal during that same light cycle. The pedestrian always has to wait an additional two light cycles to legally cross in that situation, which is an unacceptable delay that drivers don't have to deal with.
These kinds of traffic lights are set up in a way that marginalizes pedestrians in most Ontario municipalities.
It is possible with our existing signal equipment to allow the Walk signal to come on after the green has already started using a feature called a "permissive period". Basically the cutoff time for pedestrian calls is kept as late as possible, so if there's enough time left within the maximum green time to fit the Walk + Flashing Don't Walk, a pedestrian call can be served right away.
Here's an example presented by Jerry Schippa from the City of Madison, WI. He calls it "pedestrian reservice".
For minor streets, the maximum green duration is typically equal to the pedestrian crossing time, since that's already far more green time than required to clear the typical queue of vehicles. In which case regardless of how the signal is programmed, there will no longer be enough time to serve the pedestrian phase after the light has turned green. To make pedestrian reservice work, you need to increase the maximum green time by the equivalent of the desired permissive period. Jerry increases it by a couple seconds to account for the time it takes to walk to the button. After these couple seconds, the signal won't immediately respond to pedestrian calls, because there isn't enough time anymore.
For major streets, the policy in Toronto is that the signal should rest in Walk until there's actually some specific reason for the light to end (either a timed end-of-green moment or someone detected on the side street). In which case there would only be Green + Don't Walk if there is currently a TTC bus/streetcar extending the green light (and the signal therefore doesn't know exactly when the green will end). But in some other places, including York Region, signals can also rest in Green/Don't Walk, which works really well at intersections with few pedestrians. In that situation, when someone is detected on the side street, the main street signal can immediately change to yellow, whereas in Toronto the person on the side street would first need to wait for the main street pedestrian countdown. If a pedestrian is detected on the main street while the signal is resting in Green / Don't Walk, the pedestrian signal can change immediately to Walk.
There's an example of this right at the start of this video of mine (filmed at YMCA Blvd & University Blvd near Unionville GO Station).
There is some debate in the traffic engineering community about whether the Walk signal should be allowed to start after the green has already started for a movement with a permissive conflict with the crosswalk (e.g. right turns on green are permitted). The concern is that someone turning right could see a pedestrian standing still or even see the "Don't Walk" signal and assume that they can turn without yielding, but then the pedestrian gets a Walk light and steps out in front of the turning vehicle which wasn't expecting to stop. In the Netherlands, it is fairly common for signals with permissive right turns to include the constraint that the Walk or Bike signals can only start simultaneously or earlier than the parallel vehicle signal. If the signal is resting in Green/Don't Walk and a pedestrian/bike requests the main street green, the signal will first go to yellow, then red along the main street, before the ped/bike signals can start and the main street signal goes back to green.
While there is some logical justification for this constraint, I'm not convinced that it actually improves safety overall. There is a signal in a Dutch city which I programmed where there are no right turn signals along the main street, so the City required the bike and ped signals to start simultaneously or earlier than the parallel vehicle signals. No problem I thought, we'll just make the bike signal turn green every time the parallel vehicle signal turns green (like it does virtually everywhere in Toronto). But no, the City insisted that the bike signal should only turn green if there is a bike detected, because the red clearance time for bikes is (marginally) longer than the clearance time for cars. As a result, most of the time cyclists arrive and get stopped at a red light for no apparent reason. And after seeing that the parallel traffic has a green, they just ride through the red light.
Traffic signals only work when people obey them, and people only obey them when the signals seem reasonable. Making pedestrians or cyclists wait for no apparent reason will reduce signal compliance, which I suspect degrades safety more than allowing the Walk light to begin while a permissive conflict already has a green.