The Rangers captain hailed the move as a "respectful" one after van Bronckhorst opted to delay meeting his new squad until Monday in order to allow the interim management team to focus fully on the Hampden clash.
The four-man coaching team of B team manager David McCallum, his assistant Brian Gilmour, veteran player-coach Jermain Defoe and goalkeeping coach Colin Stewart have overseen the preparations while Rangers concluded a deal for their new manager.
"The gaffer didn't really want to disrupt the team this close going into the semi-final," Tavernier said.
"I am seeing him on Monday morning so I'll have a good chat with him. He didn't really want to come in and disrupt what was happening so close to the game, so really respectful and I know all the players and myself are looking forward to working with him."
Tavernier is not expecting Rangers to change their approach or style against Hibs following the departure of Steven Gerrard to Aston Villa 10 days ago.
"I don't think much changes at all," he said.
"The way we play it filters down into the B team and into the younger teams.
"We've got a great philosophy of the way we want to play and that's just filtered down the club so David and Brian have come in and it's been a seamless transition.
"They've been involved in our training sessions before through this season and last, so it's fitting so perfectly. The training hasn't really changed much, and I'm sure the tactics won't really change that much either."
Van Bronckhort is expected to be in the dug-out for the first time for the Europa League clash against Sparta Prague at Ibrox on Thursday.
"The squad we've got now is one of the strongest I've played with," Tavernier said. "Previously it might not have been that strong, going through some different situations.
"But I've got great belief and confidence in them so the new gaffer will be taking over a great squad. He'll be implementing his tactics and how he wants us to play, but the foundations are there, so I see it being a seamless transition."
James Tavernier was speaking at a Premier Sports Cup event. Premier Sports is available on Sky, Virgin TV and the Premier Player. Prices start from £12.99 per month.