The main issue for businesses and the economy is uncertainty, as the U.K. will now have to engage in a two-year negotiation process to exit the EU.
"One possible scenario is that in order to retain access to the single market, the U.K. would have to continue to accept the free movement of people from the EU. In this case little would change," Ian Dowd, director at NGA Human Resources, told CNBC via email.
"Another scenario would see restrictions of some kind on EU citizens, which could make it harder for some Europeans to work in the U.K."
Small and medium-sized businesses will likely find it harder to recruit post-Brexit, according to Eugene Mizin, CEO and co-founder of recruitment app Job Today.
"Access to talent will tighten, as fewer EU foreigners will enter Britain's workforce, which will drive up hiring costs," he told CNBC in an email. "As economic outlooks remain unclear, SMEs will look to spend less on recruiting and think twice before spending hundreds of pounds on job boards. Instead, SMEs will increasingly turn to referral and word of mouth hiring."
Other business leaders were less concerned and focused on the underlying strength of the U.K. economy.
"There will undoubtedly be short-term uncertainty in the labor market but the foundations of the U.K. economy remain strong and we remain confident that U.K. based business will continue to attract and retain talent from within U.K. and beyond its borders," Lee Rankin, managing director at recruitment firm GEM Partnership told CNBC via email.
However, rather than experiencing a brain drain of talent, the U.K. economy could become a more agile and attractive place to work.
"It depends on what path the U.K. takes – talent goes where the opportunity is, and Europe is the world's slowest growing continent, largely due to EU economic policies," Harrison Wright, managing director of Affinity Biotechnology, which specializes in biotech recruitment, told CNBC via email.
"If the U.K. takes the opportunity to make itself more attractive for business and easier for skilled workers around the world to immigrate, we will see the opposite of a brain drain."
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