It’s not a word, but perhaps it should be.
If a competitor goes after your customers by offering them faster service, all day and all night, you’ve been anytimed.
And if your boss, fearing that event, or simply trying to boost output for free, pushes you to be available all hours of the day and night, you’re being anytimed as well.
The market wants convenience and speed and price. Anytimed is a side effect of that race.