First, let's address the issue of shaving during the 3 weeks generally. According to R' YB Soloveichik, Shiurei HaRav Al Inyanei Aveilut V'Tisha B'Av, pp. 20-21, there is "a basis for shaving one's beard during this period. He contends that the observances of the three week mourning period are patterned after the twelve month period of mourning that one observes when losing a parent. A mourner during this period may shave or cut his hair when he has reached a state that his friends note his unkempt appearance. Nowadays, when people shave on a daily basis, this state is attained after a day or two. Once that state is reached, one may shave his beard and continue to shave for the rest of the mourning period. However, this leniency will only apply to the three week period [and not to the nine days leading up to Tish'ah BeAv]." - Quoted from: "The Laws of Bein HaMetzarim" [brackets added by me for clarification]
The opinion of the Rav isn't the only one out there, but he is well known for this opinion, and the following anecdote may be the source for the (now) well established, and possibly erroneous, practice among Yeshivah Bahurim to shave every Friday during Sefirah but not during the week. (By the way, since I'm recalling this from memory, if anyone sees any factual errors in my version of the anecdote, please point them out in the comments!)
When R' Aharon Lichtenstein became Rosh Yeshivah of Yeshivat Har Etzion, his father-in-law, RYB"S, told him that his opinion (which I referenced above) probably ought not apply in Israel, since it was much more acceptable there to wear a few days' growth of facial hair. Since, because of RA"L's position, he wanted to maintain a dignified look, the Rav advised him to shave Friday LiChvod Shabbath instead of daily.
Nowadays, in my own opinion, even in Israel it is much more expected that one shave daily in the workplace. However, there still may be a problem for Bahurim to shave 'Erev Shabbath, since there is no societal expectation in a Yeshivah setting that one shave in the first place, much less during Sefirah or the Three Weeks.
Now, getting to the workplace, assuming one has an expectation of looking well groomed during the week, then, borrowing from the Rav, it would seem it is appropriate to shave on Friday as well, just as one might if, Lo 'Aleinu, one were in the 12 month mourning period for a parent.
Unfortunately, the link at YU Torah to the Shi'ur where I found the quote is dead. If I find a new link online for the Shi'ur or for the quote, I will add it.
The link now works. I will keep a copy of the Shi'ur on hand. If I can find a new site to host it, perhaps a blog, I will provide that link in a future version of this answer.