Lots to consider. I recently retired as an IT Manager for a small town. Our situation and needs were similar as we streamed board meetings to the public, setup conferencing and training rooms, etc. As did most other municipalities, we used specialized AV companies for much of this type of work - especially for setting up the board room and steaming. The company we used did lots of work for churches also.

You can easily find products that will claim to accomplish most anything you want to do. But knowing which ones actually work and deliver the quality and reliability acceptable for your situation and organization requires direct experience and/or trial and error. The specialist's experience is hard to beat. If you choose to do your work in-house, beware of inexpensive video processors and converters. We found they never delivered the quality we needed. And higher priced "pro" equipment sometimes failed to deliver also.

Having said that - I would assume the large monitors at the front of the church need to be real time - with no perceptible delay at all and tied into an online streaming system? While the ones in the other locations may not need to be?

For conference rooms and remote locations, we purchased micro computers ourselves and connected them to large wall mounted TVs (as opposed to buying commercial grade large monitors and specialized systems.). This is relatively inexpensive, works pretty well, and gives you a lot of flexibility and functionality. You can stream from different sources, conference, etc. If using Windows 10 Pro, you can use remote desktop when needed. A good wireless mouse and keyboard will work a substantial distance to control everything. Maybe this would work for your nursery and fellowship hall?

I would guess you already stream your services over the Internet, or are thinking to do that with the upgraded system? We used home-brew solutions for years that worked pretty well. And there are some streaming services that can be really inexpensive for small organizations - allowing you to pay for blocks of bandwidth/data up front and only charge for what you use. We used dacast.com for many years. (I just checked the Town's website and seems they are still using them.)

Hope some of this rambling is useful!