How to make the image on BillboardGui change?

The problem is because the Left and Right variables are set to the property's value but when changed it doesn't update the actual property due to how scripting/coding works.

So instead do this:

local Left = Overheads:WaitForChild("Left")
local Right = Overheads:WaitForChild("Right")
if Stats == "God" then

	----- THE SCRIPT BELOW WON'T CHANGE THE IMAGE
	Left.Image = "rbxassetid://11260911548" --- This is inside of the overhead
	Right.Image = "rbxassetid://11260917532" ---This is inside of the overhead
	---- EVERYTHING BELOW WORKS NORMALLY
end

Full script after fix:

local ReplicatedStorage = game:GetService("ReplicatedStorage")
local MarketplaceService = game:GetService("MarketplaceService")
local Players = game:GetService("Players")

local Overheads = ReplicatedStorage:WaitForChild("OverheadsGui")
local IDVIP = 91491857      ----//VIP ID
local GroupId = 11540689    ----// GROUP ID

Players.PlayerAdded:Connect(function(Player)
	Player.CharacterAdded:Connect(function(Character)
		-- These variables are used regardless of whether the user owns the gamepass or not
		-- so we can define them outside the if statement
		local Humanoid = Character:WaitForChild("Humanoid")
		local Head = Character:WaitForChild("Head")

		local Stats = Player:WaitForChild("leaderstats").Rank.Value

		-- Again, this happens regardless of the if statement
		Humanoid.DisplayDistanceType = Enum.HumanoidDisplayDistanceType.None
		if
			MarketplaceService:UserOwnsGamePassAsync(Player.UserId, IDVIP) then
			-- Always indent blocks of code
			local Rank = "VIP"

			local Left = Overheads:WaitForChild("Left")
			local Right = Overheads:WaitForChild("Right")
			if Stats == "God" then

				----- THE SCRIPT BELOW WON'T CHANGE THE IMAGE
				Left.Image = "rbxassetid://11260911548" --- This is inside of the overhead
				Right.Image = "rbxassetid://11260917532" ---This is inside of the overhead
				---- EVERYTHING BELOW WORKS NORMALLY
			end

			local CloneGui = Overheads:Clone()
			CloneGui.Username.Text = Player.Name
			CloneGui.Rank.Text = Rank
			CloneGui.Parent = Head
			CloneGui.Name = "OverheadGui"
			local Role = "Guest"
			pcall(function()
				Role = Player:GetRoleInGroup(GroupId)
			end)
			CloneGui.Role.Text = Role
		else
			local Left = Overheads.Left.Image
			local Right = Overheads.Right.Image
			-- Always indent blocks of code
			local CloneGui = Overheads:Clone()
			CloneGui.Username.Text = Player.Name
			CloneGui.Name = "OverheadGui"
			CloneGui.Parent = Head
			local Role = "Guest"
			pcall(function()
				Role = Player:GetRoleInGroup(GroupId)
			end)
			CloneGui.Role.Text = Role
		end
	end)
end)