Remote Event, Cooldown won't work, why?

I think your problem is that your using OnServerEvent ( which is an entirely different code block ) meaning that the wait WONT transfer over, to fix this, do OnServerEvent outside of your .MouseClick function and make a variable for debounce.

like so:

local part = script.Parent
local clickd = script.Parent.ClickDetector

local berrysLeft = script.Parent.BerryLeft
local berryAmount = script.Parent.BerryAmount

local rep = game:GetService("ReplicatedStorage")
local remote = rep.Remotes.ClickBerrys

local deBerry = 1

local debounce = false
local DebWait = 1 -- set a default of one

local getClickRemote = rep.TestRemote.ChangeValue
local getClickValueRemote = rep.TestRemote.getChangedValue

getClickValueRemote.OnServerEvent:Connect(function(player, CollectingSpeedV)
   DebWait = CollectingSpeedV
   print("The current debounce is: " .. DebWait) -- for debugging
end)

game.Players.PlayerAdded:Connect(function(player)
    player.CharacterAdded:Connect(function(char)

        local CollectingSpeed = player.Skills.CollectingSpeed

        CollectingSpeed.Changed:Connect(function(changed)
            print(CollectingSpeed.Value)
        end)

        clickd.MouseClick:Connect(function(clicked)

            if not debounce then
                debounce = true

                berrysLeft.Value = berrysLeft.Value - deBerry
                part.Size = part.Size + Vector3.new(0,-0.25,0)

                wait(DebWait)
                debounce = false
            end
        end)
    end)
end)