There will certainly be some jitter, but it'll be like rush-hour traffic. Sure, some people get in a little earlier or later and they leave from different points, but the result is that we still have traffic that peaks. I don't think the issue is that everyone is turning it on at the same time.
For example, let's say that people have dumb thermostats and they turn them down 2 degrees overnight - because you don't want to turn it down too aggressively since it won't warm the place before you wake. They wake up and turn them up 2 degrees. There's jitter so it's not all at once. The home heats up over the next 20 minutes and the heating turns off. Let's assume that everyone wakes up evenly distributed between 6am and 7am. The grid is handling 1/3rd of customers (or less) at any given time. By minute 20, 1/3rd have turned on their heat, but by minute 21 the first 1/60th of customers have their systems turn off.
By contrast, let's say that people have smart thermostats and they turn them down 6 degrees overnight. The thermostats likewise introduce jitter and we'll say they evenly distribute over a 1 hour period. Given the greater amount that they need to warm the place, they'll each be on longer - let's say for an hour even though degrees aren't linear. Half of them start in the first half of the hour so by minute 30, you're dealing with half of the customers running their heat - but because there's longer to go to heat up a place 6 degrees, none of them are shutting off. At minute 31, now 31/60ths of customers are running their heat. Then it's 32/60ths. This continues until we're at all customers with the heat on simultaneously. We'll be at 50/60ths usage for 20 minutes which is 2.5x more load than the other example.
This principle holds regardless of different building properties or increased jitter. A greater percentage of people will have their heating systems on at any given time in the morning due to this.
This is important because it means that it increases the likelihood that utilities need to run so-called "peaker plants" that are more costly to operate and pollute more to handle the load.
Sometimes utility companies offer incentives around avoiding this problem. I believe some California utilities have incentives for charging your electric car overnight when the load on the grid is minimal - because it's cheaper for the electric company to supply electricity then. Likewise, it would be cheaper for the electric company to supply electricity for heating then. Some electric companies have off-peak rates. Some electric companies have incentive programs where you can enroll your smart thermostat with them and they can shut off your AC for an hour when they're trying to shed load (you can override it, but it can help them shed load from lots of people who don't notice a degree or two change).
No, everyone won't be turning on their heating at the same time. However, if everyones smart thermostats run the heating system for 2 hours in the morning rather than 20 minutes in the morning because they more aggressively manage the overnight temperature, there's going to be a lot more overlapping running. This usage will overlap with other peak-hour usage like hot water heaters, toasters, microwaves, stoves, and businesses opening up - compared with overnight heat usage when businesses are closed, people aren't showering or making food, etc.
The answer is probably smarter smart thermostats. They turn themselves down overnight, but then the electric company manages the overnight temperature so that they don't face a morning rush to heat and have to use peaker plants. This would be done with consent of the users and within ranges specified by the users and to the user's benefit in terms of saving money - ie. the person wouldn't have to pay more just because their heat was kept a bit higher overnight and it wouldn't be kept above their comfort level. Likewise, users could opt into a program where they might be able to save money (say $5-10/mo) with the note that during peak times their home might heat up more slowly than it otherwise would - if it would normally take an hour to go from 60F to 68F, maybe it takes 1.5 or 1.75 hours. Again, that would allow the utility company to shed load while not inconveniencing consumers too much and providing adequate compensation for what is an inconvenience.
Electrifying our heating can mean that our heat starts coming from sources like wind and solar rather than gas and oil. However, it'll also provide some challenges around peak-time usage. These aren't insurmountable problems, but they do exist.