‘Any Jew’ surely means any Jew, does it not? Attempts to refine an inclusive principle into an exclusive one, seem to run contrary to the original intention. That much appears unequivocal, to this Gentile observer.
I don't really know how things are in Britain, but I will try to give an American parallel: American lawmakers and other individuals involved in the legal world often debate the issue of "what did the Founding Fathers mean when they said X" or "what did the Founding Fathers mean when they including in the Constitution Y". This is similar: The question being asked here is: What did the first government of Israel mean when they said they would allow all
Jews to come to Israel. I.e., "who is a Jew?"
Now, besides that, the law as it is contains a clause called in Hebrew סעיף הנכד - Se'if Ha'neched, or the Grandchild Clause. This allows descendants of Jews (sons/daughters and grandchildren (but not further down)) who are not Jews themselves (again, whatever that means) to also make aliyah, i.e., immigrate to Israel and receive citizenship. When the state was created, 99.99% of the people who were interested in immigrating and becoming Israeli citizens were Jews who could also claim and generally provide documentary and testamental proof of being descendants of Jews on both sides, so the above-mentioned clause was not a problem in terms of preserving the Jewish identity and majority of the country.
However, since then, Israel has managed to become a successful country in many terms, and now many non-Jews have an interest in immigrating to Israel. That is exactly what happened when the Iron Curtain went down. Hundreds of thousands of former USSR citizens looking for a better life swarmed Israel. Many of them had only a Jewish father or Jewish grandfather and did not even identify as Jews in the slightest, which, from what I gather, was considered problematic even in the eyes of Reform and Conservative Judaism of the time.
A recently conducted study has revealed that there are some 4.8 million people worldwide today who could attain citizenship in Israel via this clause, while not having even the slightest connection to any form of Judaism. This, of course, poses a major danger to the Jewish identity of the country, which is the reason it was even founded.
These, essentially, are the main issues that are being debated. There are also some minor ones, but I won't get into those now.