It's also a zoo and situated in the capital so under Zorro's definition it's as valid as Regent's Park for the title.
IMO a national zoo needs some government funding and official recognition to be considered nations. As has been stated, London Zoo ceased to be the national zoo 40 years or so ago.
Look at it any way you wish but at the end of the day London zoo is the National zoo of England did the Chinese give the pair of Giant Pandas in 1973 to Battersea park zoo? NO, they went to London zoo, did the Chinese give another pair of Pandas to any other zoo in the USA in the 1970s to any other than Washinton zoo DC, No, but believe what ever you wish, "Iam good with that"