The repeal of ordinances pursuant to Section 51947 shall not apply to:

1. (a) Ordinances under which vested rights have accrued to the extent legally necessary to protect said vested rights.
2. (b) Ordinances relating to proceedings for street or other public improvements, to the extent legally necessary to preserve said proceedings completed or pending.

3. (c) Proceedings for opening, extending, widening, straightening, or changing the grade of streets or other public places.

   These proceedings shall be continued and conducted by the city-county, with the same effect as if continued and conducted by the city or county which commenced them.