The time for tfila in the morning is, a priori (lehathila), up to the fourth halachic hour (a third of the daytime) and, after the fact if you missed the time, up to halachic noon (it is an explicit Mishna in Brakhot 4:1).
Therefore, if after the fact, you are at risk of missing the zman tfila, you should skip the parts mentioned in the artscroll siddur.
There is a question whether to skip shema in such a scenario and you should ask your rav as there are different opinions (some believe the order of prayer should not be changed). See here (mid-page) for one possible approach
You must say the shema no later than the end of the third halachic
hour of the day in the morning. If you did not say it in time:
- If it is before the fourth halachic hour of the day, say the entire shacharit service, as long as you will complete the amida before the
end of the fourth halachic hour.
- If it is after the fourth halachic hour of the day but before the end of the sixth halachic hour of the day, say birchot ha'shachar (and
you may also say korbanot) and then skip all of shacharit up to the
amida, which you should say as soon as possible.
And next time you should set an alarm clock :->