Well every day is a new day Assassinated and you treat it as such and start afresh each morning. There will of course be days when nothing goes according to plan - that happens whether you follow a schedule or not - you have to be able to brush those days off and start afresh the following day.
In terms of within the individual day, if my DC had wanted to sleep when it wasn't nap time, then I worked hard to keep them awake - not always possible, but it worked most of the time. It was hard work for sure, and like I said - those first few months are by far the hardest - but you get on with it. If it's the other way around and they didn't sleep at all when they were meant to, then yes, you get them up at the scheduled time, even if they have only slept for 10 min and carry on as best you can.
You do of course have to have a bit of wisdom to deal with those situations and you read your baby, so you might bring forward bed time that night by 10/15 min to help make up for the lost nap time earlier, but you pretty much try to stick to schedule and deal with anything that goes wrong as it comes up.
I mentioned above that my first dc was a total speed-feeder and so I always had way more time to fill than the book suggested because feeds only took 15-20min. I worked around that.
I did the same with my youngest two who took a hell of a lot longer to feed because there were two of them and because of the reflux which resulted in so many throw-ups and clean-ups that feeds seemed to take forever! Again - we worked around it.
Same with middle DC who also had severe reflux, but also severe eczema so every nappy change/clothes change took way longer than it would normally because of all the different creams etc - we worked around it.
Then you get up at 7am the next day and start all over again no matter how the previous day or night went. Each day is a new beginning.
That's not trying to be defensive at all by the way - just trying to explain it as it worked for us.