Quoted:
How would you handle being told your job was over after next week and you get another job offer and you can start before the end of next week. What do you tell the new employer about why you didn't give the customary 2 weeks notice so they don't think you are going to just up and leave on them?
ETA - A new company is taking over the old job and bringing in their own people.
So... your boss tells you you aren't going to have a job any more, you get another offer, and think there'll be a problem by taking it?
Employers do not call past jobs and ask if you gave notice or kept to it. They only know what you tell them.
In this case I'd tell the new employer, "I can start immediately if you like, because the old job is going away." They'll understand that.
If you really want to finish out the old job, then tell the new employer you can start in two weeks.