We had a big job "on" today!

Yes, it means 'scheduled'. It's very common in British English.

It's not an adverb, by the way.
 
It's a preposition. You don't need to worry about this. It doesn't help you in any way how it's classed. You need to know what language means and how it's used. This is the grammatical pattern:

have something on
 
Yes, it means 'scheduled'. It's very common in British English.

It's not an adverb, by the way.

"On today" is common in BrE?

In AmE, we would just say "we had a big job today." "On today" is an error.
 
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