Someone with a background in agriculture, biology or biotechnology can design the basic process and determine the principle mass and energy flows - how much area, how much artificial lighting if any, how much water, air changes if aneclosed building, nutrient or fertilizer dosing etc.
The basic design stage should allready consider the logistics of your plant, but also things like workplace health and safety
Civil and structural engineers should design the main vessels and the general plant layout
There are a number of works you specified that would be their own mechanical engineering speciality: HVAC but mostly processing and packaging equipment. Others, like plumbing/pumping can probalby be done by most MEs with some experience in plant design
Electrical and control systems are on or two additional engineers you need.
In theory, the process designer designs the process, then the civil engineers size and place the vessels in their drawings, then the ME figure out the plumbing and at last the MEs show up to design the electrical systems for the machinery. In actuality, you will iterate through this process or some steps of it a few times.
This is about a dozen specialities you need. However, sometimes you can buy expertise along with equipment, this migh be the case with processing or packaging equipment or dosing stations for nutrients. You still need someone on your team with a basic understating to talk to them, however!
For example, if I wanted to setup such a facility, who would I have to consult?
You either find a consulting engineering firm with a lot of experience in designing and planning (and building!) such a plant. Or you find anexperienced hydroponics expert (the first bullet point) and a consulting firm with experience in a relevant field like wastewater.
Alternativly, you find a company specialized in building and selling hydroponics farms. This will give you less choice over the final plant - the company will want to work with their preferred components and concepts, and crucially they will want to reuse as much egnineering work from previous projects as they can.