After a town becomes a city under this chapter, the costs and expenses of the circuit court for the county, including jury costs, and the salaries of the judge and clerk of the circuit court and the clerk, attorney for the Commonwealth and sheriff of the county shall be borne by the city and county in the proportion that the population of each bears to the aggregate population of the city and county. Such expenses and costs shall include stationery, furniture, books, office supplies and equipment for the court and clerk’s office; supplies, repairs and alterations on the buildings used jointly by the city and county; and insurance, fuel, water, lights, etc., used in and about the buildings and the grounds thereto. The cost of any new building erected for the joint use of the city and county shall be provided for in like manner. However, in the case of buildings used jointly by the City of Covington and Alleghany County, no repairs or alterations shall be made to any such building, and no new building shall be erected without the approval of the governing body of both the city and the county. If such governing bodies cannot agree, relevant controversies shall be resolved in the manner provided by § 15.2-3829.