I read that anyone who pays the fee (currently 30 pounds) can view the register (including the names of the godparents) of any baptism conducted in the Church of England.
In this case,
the baptism is taking place at the private family chapel inside Windsor Castle (as I suspected it might). I'm also not surprised that Archie's godparents prefer not to be announced. Chris Ship and other reporters have been told by their sources that the godparents who were chosen are all private citizens and friends of Harry & Meghan. It is possible to guess who some of them might be: I believe Genevieve Hillis and Lindsay Roth, both of whom are college friends of Meghan's were probably chosen. Both ladies were with Meghan at Wimbledon yesterday:
https://pbs.twimg.com/media/D-pTMJBWsAU_Lpn.jpg:large Hillis is on the left; Roth is on the right
I also think it's possible that Meghan's friend, Markus Anderson of Soho House, was chosen. On Harry's side, possibly one of the van Straubenzee brothers, Mark Dyer (Harry's mentor), and perhaps one of the Spencer cousins or one of Diana's siblings were selected. I think the criticism and fuss over no godparent announcement and no filming of arrivals is ridiculous.
A number of Meghan's friends and relatives have been harassed from day one by the media, and by haters on the Internet. A British woman who was erroneously suspected to be serving as a doula for Meghan was hounded by paparazzi and bombarded by phone calls to her unlisted home number, and she's never even met Meghan. A medical doctor who was once associated with Meghan in California was tracked down and harassed by the media. The Sussexes have also received death threats. A perpetrator was arrested by Scotland Yard. That story has not been widely disseminated. It's understandable that Archie's godparents in the current OTT climate do not wish to be formally identified.
BTW, all royal christenings are 'private,' but those in immediate direct line to the throne who are working royals generally have some details released, along with photos. The public will see official photos of the Sussexes, Wales grandparents, and the Cambridges, et al, taken after Archie's christening. I don't see why that isn't sufficient. The press apparently would like the ability to have at least pool photographers and reporters set up to record arrivals, but that can't happen anyway, as the christening is taking place inside Windsor Castle's private chapel.
As far as the press being able to find out by seeking the church register, that won't be as easy as some people seem to think.
The Windsor Castle chapel is private, and St. George's chapel at Windsor is a royal peculiar* under the Queen's jurisdiction. Any attempt to view official Windsor chapel or St. George's chapel registers would have to be approved by Archie's paternal great-grandmother.

I don't see that ever happening. Not that it isn't possible to guess some of the godparents' identities or to eventually find out some information via leaks.
*royal peculiar =
"a Church of England parish or church exempt from the jurisdiction of the diocese... and subject to the direct jurisdiction of the monarch."
Those who think Meghan and Harry are unnecessarily making announcements, think again. M&H have said nothing directly. BP has released an official announcement about the christening, and some of the media were alerted in advance, but not by the Sussexes.

The press often make inquiries and royal communication staff are accustomed to providing official information about the activities of the royals when deemed necessary. For example, Zara Tindall was said to be upset when media intruded and took pictures of arrivals at her second daughter's christening earlier this year. Neither Zara nor her family are working members of the royal family and no official announcements were made of any kind about her daughter's christening.
Of course Archie's christening would be addressed beforehand since Harry & Meghan are working royals, and they are willing to share some details about Archie for those who bear goodwill. But make no mistake that Prince Harry is trying to protect his family from undue intrusion, and he seems to want Archie to not have to face the kind of scrutiny he grew up dealing with. IMO, the way Meghan is constantly criticized in some quarters of the media only makes Harry less amenable to being forthcoming.
Both Harry and William have been impacted by what happened to their mother. They were unable to protect her from the media, so they will move heaven and earth to protect their families. In Harry's case, I think it's obvious that right now he is sending the message that his son will be raised in as private a way as possible. The Cambridge children will be working royals, so how things were handled at their christenings is different. I think Meghan is in agreement with Harry's wishes, but that she would probably be a bit more relaxed about sharing, if not for the OTT negativity and the fact of a 24/7 royal fishbowl being a drawback to being more open. Perhaps we will get to see something of Frogmore Cottage, the dogs, and Archie in the September issue of British
Vogue, which Meghan is said to be guest-editing. But there's no guarantee. In any case, no matter what the Sussexes do or don't do, they will be criticized.
Some royal reporters want to have their cake and eat it too. The media's narrative has been how far Harry and Archie are from the throne which makes them 'less important,' which is not far from being accurate, particularly as time goes on and the Cambridge children come of age. It's just that the tone of such media comments is always negative and overly contradictory. The issue right now is that the Sussexes are popular and there's a lot of interest in them (well meaning and negative), and so stories about them (whether made-up or legitimate) are lucrative for the media and paparazzi.