The Angelus is a way of marking the hours of the day for the laity: 6 am, noon, 6 pm. So it is sort of analogous to the LOTH in that sense; and it was meant to be a prayer easily memorized so you could drop to your knees and pray it wherever you were, such as working in the fields. While it is not formally attached to the LOTH in any way, the monks at our abbey do pray the Angelus three times per day: at around 6 am after Vigils (Matins) is completed, at about 12:10 pm at the end of Sext, and at 8 pm, at the end of Compline.
There is no real connection between Vatican II and the Angelus. The Angelus is an ancient prayer and it is still said regularly. The church in the town closest to me still rings the Angelus three times per day.
I usually say it myself after Vigils (5:30-6 am), Sext (12:10 pm) and Compline (whenever dinner finishes), It’s not an obligatory prayer so there are times, either when rushed or too tired that I omit it.