#K4, 40 bytes
Solution:
{+/1<.q.mod[d+!(."."/:|"."\:x)-d:.z.d]7}
Explanation:
Calculate the difference between the dates, use modulo 7 to ignore weekends, sum up.
{+/1<.q.mod[d+!(."."/:|"."\:x)-d:.z.d]7} / the solution
.q.mod[ ]7 / modulo with 7
.z.d / UTC date
d: / save as d
- / subtract from
( ) / do this together
"."\:x / split input on "."
| / reverse
"."/: / join back with "."
. / take the value
! / range 0..the difference
d+ / add today's date to range
1< / is 1 less than the modulo (ie is this mon-fri)?
+/ / sum up
Notes:
- same byte alternative to the date parsing:
"D",ドル/|"."\:x