Dealing with all the different stakeholders for data and analytics can feel like herding cats. How can we be successful in managing their competing priorities and goals?

The only way to effectively manage priorities is to have some understanding of the expected business benefits of every significant request to the data team.  And by a business benefit, this means either increased revenue, reduced cost, or managed risk.  

Here is my perspective on this
Clearly Define Objectives: Start by establishing clear and measurable objectives for your data and analytics projects. Ensure that all stakeholders understand these objectives and agree upon them. This will serve as a common goal that can help align their priorities.
Clearly defined objectives provide a clear direction for stakeholders to follow.
They help in aligning competing priorities by emphasising a shared goal.
Regularly review and update objectives to ensure they remain relevant.
Prioritise and Negotiate: Inevitably, stakeholders will have competing priorities. Work with them to prioritise these priorities, considering the potential impact and urgency of each. Engage in negotiation to find compromises when necessary.
Prioritisation helps in allocating resources and efforts efficiently.
Negotiation can resolve conflicts and find middle ground among stakeholders.
It ensures that the most critical aspects of the project are addressed first.
Establish a Steering Committee: Create a cross-functional steering committee with representation from different stakeholder groups. This committee can help make decisions, resolve conflicts, and provide guidance for data and analytics projects.
A steering committee brings diverse perspectives to project decision-making.
It helps in reaching consensus and managing conflicts at a higher level.
The committee can provide strategic direction and oversight for projects.

I would say the most important thing is that the central data governance team should not be responsible for prioritizing work. I tend to introduce a role called data owner: senior stakeholders accountable for one or more data sets. I get them to sit on a data governance committee or a council, and I get them to set the direction and prioritize activities.

People respond much better if they feel they've had a say in it. It's all about empowering the business to make those decisions themselves and just supporting them as the data governance team in that, rather than trying to do that for them.