Given that there is absolutely no law that prohibits people from photographing her once she's in public,
But there is.
The conduct of paparazzi photographers might amount to harrassment under the Protection from Harrassment Act 1997, which gives a statutory cause of action. That is in addition to any other possible claims regarding breach of confidence, nuisance, misuse of private information, invasion of privacy, trespass, etc., which have tended to arise in similar situations (though the question here only posits that photos were taken in public places).
Even though the Princess of Wales is famous, she has the benefit of Article 8 of the European Convention on Human Rights, which secures "the right to respect for [her] private and family life, [her] home and [her] correspondence." (A paparazzo is not a public authority, but a court being asked to adjudicate a civil claim is.) In Von Hannover v Germany 59320/00 [2004] ECHR 294, the European Court ruled in favour of Princess Caroline of Monaco, regarding magazine coverage of her daily activities like shopping. The Bundesverfassungsgericht in Germany had said that as a figure of contemporary society who was not in a secluded place, her right to privacy did not defeat the right of freedom of the press under Article 10. On the contrary, the ECtHR said that:
the publication of the photos and articles in question, of which the sole purpose was to satisfy the curiosity of a particular readership regarding the details of the applicant's private life, cannot be deemed to contribute to any debate of general interest to society despite the applicant being known to the public.
They said that this situation was different from intrusion into the life of a politician, where there could be genuine public interest in their private conduct; although Princess Caroline was a senior member of Monaco's royal family, who sometimes represented that family at charitable or cultural events, she did not hold any office of state. The "climate of continual harrassment" by paparazzi was part of this assessment.
Now, the Princess of Wales is a member of the royal family, but doesn't run the country, and unlike Caroline is not even in the line of succession. If she is just out shopping, then her conduct is not inherently scandalous. (In contrast, Campbell v MGN [2004] UKHL 22 engaged questions of "where a public figure chooses to present a false image and make untrue pronouncements about his or her life", in that instance using drugs while claiming to not use them.) Although it's not taking place in a private location, von Hannover shows that consideration is not enough in itself to exclude the application of her Article 8 rights.
There is still a balancing exercise involved, since (quoting Lord Mance's summary from PJS v NGN [2016] UKSC 26 at para 20)
(i) neither article has preference over the other, (ii) where their values are in conflict, what is necessary is an intense focus on the comparative importance of the rights being claimed in the individual case, (iii) the justifications for interfering with or restricting each right must be taken into account and (iv) the proportionality test must be applied.
In other words, while the outcome of any suit would depend on the actual facts, the Princess's Article 8 rights are quite high (even though she's a celebrity in a public place), and the protection afforded to mere celebrity gossip coverage under Article 10 is quite low.
