The fear among some is that outside of the EU the regulations that have helped protect the environment will be weakened.
The Birds Directive, for example, seeks to protect 500 wild bird species in the EU, while the Habitats Directive seeks to ensure the conservation of "rare, threatened or endemic animal and plant species."
A spokesperson for WWF-U.K. told CNBC that a Brexit "undeniably presents a number of risks to our environment."
The spokesperson went on to add that it was vital that the U.K.'s new leadership – whoever that may be – "understands that people want it to build on, rather than tear down, the EU standards that have helped get our seas and rivers into good health, promoted recycling, fought climate change and protected rare species."
In its financial year ending on June 30, 2015, WWF-U.K. says it received £907,000 ($1.21 million) in funding from the European Union to, among other things, work on protecting our rivers and fight climate change.
The Royal Society for the Protection of Birds is the largest nature conservation charity in the U.K., and has also received funding from the EU for projects it is involved in.
"It is essential that we do not lose the current, hard won, level of legal protection," Mike Clarke, its chief executive, said in a statement.
"Given the current state of nature, we should be looking to improve the implementation of existing legal protection and, where necessary, to increase it," Clarke added.
The CEO of Friends of the Earth - which says that for the financial year of 2014-15, less than one percent of its funding came from the EU - said in an online article on the group's website that last Friday was the beginning of a "fight" to ensure the U.K. did not "water down environmental safeguards we've inherited from the European Union."
Craig Bennett went on to state that Friends of the Earth's priority would be to "ensure that every MP understands loud and clear that the referendum result was not a mandate to weaken our environmental protection."
For its part, the WWF-U.K. spokesperson added that, "Given the extent of the threats facing the natural world, and the huge costs we'll all face if they are not addressed, pessimism and inaction are not sensible options." The organization would therefore be speaking to ministers and making the case for a greener economy and "better stewardship of our unique wild places."