This is challenging indeed and something I've had to deal with at every company I've worked for. What I've fund helps keep me and the business teams sain is to plan to launch features 14 days after the official planned released date. This makes product nervous most of the time, but most of the time they're also delayed so it all works out in the end.
To establish credibility with a new team, the first step is understanding the team's need, laying out a vision for how you can best add value, and aligning around expectations. It is important to know the user, the market, and the product so that you can engage with the cross-functional team in a meaningful way from day one. With a clear set of objectives and foundational understanding of the space, you can quickly begin to make an impact on the team.
Goes back to the shared goals - which at a high level, are hard to argue with - revenue, cost savings, customer success, etc. Once you get that common agreement, then it's about the strategy / the "how" to get there. If there are disagreements here, I would start with trying to understand why and seeing it from both of their vantage points. Then trying to see if you can get them 1:1 to understand the other point of view or better yet, get them to talk to each other. Ultimately though if all that doesn't work, you may need to get a tie breaker that's someone else and who they will listen to.