Put on safety glasses and make a horizontal cut along the back of the old wood window sill using a circular saw. Be sure to set the blade to cut the thickness of the old sill.
Make two vertical cuts along the two ends of the old wood window sill with an oscillating multi-function tool. Then, make one more vertical cut in the center of the sill.
Insert a pry bar under the vertical cut you made in the middle of the old sill and pry up, breaking the old sill out of the window.
Pull the old sill out and dispose of it.
Measure the space in the window where the new sill will go, then measure, mark and cut a new wood sill to fit the space using a circular saw.
Fit the new sill into the empty space by hand and tap it in tight against the window using a hammer and a piece of scrap lumber. Hold the scrap lumber against the new sill and strike the scrap with a hammer so you don't mar the new sill.
Measure and cut a new sill nose to fit in front of the replacement sill with a tape measure and circular saw.
Fit the sill nose in front of the new sill by hand, then tap it in with a hammer and scrap lumber like you did with the sill.
Fasten the sill nose to the replacement sill with stainless steel nails and a nail gun. Then spread a bead of silicone caulk around the new sill and nose using a caulk gun.
Apply a coat of wood sealer to the new sill and nose with wood sealers using a paintbrush. Allow the sealer to dry, then prime and paint the sill to match the window.