So, if the veto would have been overridden anyway, military funding was never in jeopardy and Trump should not have signed it.
Trump will remember this budget the next time.
The President would have to hold the bill 10 days and it would become law. If congress was adjourned then it would be pocket veto. With congress in session, yes it would be law just like if he signed it. Problem being the govt. would have shut down at midnite and the dems would have a heyday with a 10 day Trump caused shutdown.