No way to confidently tell if Harry will successfully argue that News UK does not have a limitation defence. It's very fact-specific, and the particulars of Harry's case aren't available.
Having said that, there is a judgment available from a hearing last year against the Mirror Group, where the Mirror Group tried exactly the same argument about the limitation period to try to get the phone hacking claims dismissed, and the approach that the judge took to deciding that might indicate what the judge in Harry's case will be thinking.
In the Mirror case, the judge decided that if you have had your phone hacked (or otherwise been unlawfully surveilled), and then an article has been published containing information from that hacking, you have suffered from two different instances of misuse of your private information: first, when your phone was hacked, and second when the article was published. As these are two separate causes of action, the 6 year limitation period for each runs separately.
As in Harry's case, the Mirror tried to argue that it had been more than 6 years since both the hacking and the publication, so all claims were time-barred. The claimants argued that the Mirror Group had "deliberately concealed" the facts so that they couldn't bring any of these claims in time.
The judge found that, in respect of the publication claims, there was inherently no deliberate concealment of facts, because the articles were public and the claimants knew about them at the time. So all publication claims had to be brought within 6 years of the articles actually being published. As none of them had, the claims for misuse of private information through publication were time-barred and dismissed by the court.
For the claims about the actual phone hacking, it was less clear cut. The Mirror Group had certainly deliberately concealed the facts about their hacking activities - they falsely attributed their articles to "close friends" and "sources" to put people off the scent. The judge accepted that none of the claimants had actually known that they had been hacked until about 2015, when their friends had brought successful claims and told them to do the same. The judge therefore had to look at "constructive knowledge" - ie, would a reasonable person have twigged that they were being hacked, even if the claimants themselves did not? If so, the 6 year limitation period would have started running from then.
The judge didn't really come to a conclusion on this. These pre-trial hearings are for obvious issues that don't need a full trial to be decided, and if it's not obvious, it needs to continue to be argued about at trial. So that's what is happening with the Mirror case - the trial will hear the full issue of whether the claimants should have known about the phone hacking earlier than they actually did, and will therefore determine when the 6 year limitation period runs from.
So to apply this to Harry's case - he presumably also has both instances where he has been hacked/surveilled, and instances where articles have been published about what was found. If his judge follows the other judge's reasoning, there will be no waiving of the 6 year limitation period for the publication claims. They will all be stuck out if they weren't brought in time.
For the phone hacking/surveillance claims in themselves, the 6 years will run from whenever Harry either actually knew he was a victim, or should have reasonably known. As with the Mirror judge, Harry's judge might find that constructive knowledge is a trial point, and let the claim (and the limitation defence) run to a full trial. However, although we are lacking on the facts, it does seem that Harry has a bigger hurdle to overcome in terms of constructive knowledge than the Mirror claimants did. They were B-list celebrities, and friends of B-list celebrities. It was pretty reasonable of them not to think that they were not Hollywood enough to be phone-hacked. Harry, however, is an extremely high-profile figure. His own brother, who has a similar level of profile, kicked off the entire phone hacking investigation. And Harry was aware in at least 2012 that there was sufficient nefarious activity from News UK to require a "secret deal" in respect of the claims. These same factors don't apply to Hugh Grant in the same way - he's neither B-list nor royalty and didn't have a secret deal - so perhaps the judge will decide differently for Hugh and Harry as to when they should have been aware of the possibility of phone hacking. We shall see!