Digging quarry timelapse

Retina 0.8.2, 163 156 bytes

.+
$*_
(_+)¶(_+)
$2¶$1¶$1
r`__\G
$%`$&
¶
;
(?<=(_+);.*)(?<=;_+;\1*)_
¶$`_
m`^_+;
__
+`(>*)_(_+)(_+<?;)\2
$1_$.2$* $3¶$1>$2<;
T`>\_` \\`>+_
T`\\\_;<`V/_`.<|;

Try it online! Explanation:

.+
$*_

Convert the inputs to unary. This gives us W¶S.

(_+)¶(_+)
$2¶$1¶$1

Swap the inputs and duplicate the width. This gives us S¶W¶W.

r`__\G
$%`$&

Calculate the volume of the quarry. This gives us S¶W¶V.

¶
;

Join the inputs into one line. This gives us S;W;V.

(?<=(_+);.*)(?<=;_+;\1*)_
¶$`_

Calculate the amount of progress for each day on its own line. Each day has the format S;W;D, where D is 0 on the first line and increments by S each day until it reaches V.

m`^_+;
__

Delete S and increase W by 2 on each line. This gives us G;D for each day.

+`(>*)_(_+)(_+<?;)\2
$1_$.2$* $3¶$1>$2<;

While D is nonzero, dig either D or G-2 from the line (so the first and last characters are always left), moving the depth to the next line. Each line is indented with one more > than the previous. Newly dug lines also include a <.

T`>\_` \\`>+_

Turn the indent into spaces and the following _ into a \.

T`\\\_;<`V/_`.<|;

If a < is following a \ then turn it into a V, if it's following a _ then turn it into a /. Delete all the <s and ;s.