Is it possible to do Bergen-Sognefjord-Naeroyfjord-Bergen in one day?

I am not sure, but I'd guess it is possible. I think the Bergen-Flam express boat only runs once a day in each direction, departing in the morning, and it's not a quick trip (5 hr. 40 min.?). I'd suggest researching the schedule, which is probably available on norwaysbest.com among other websites.

In peak season I think there are six round-trip ferries between Gudvangen and Flam, but some of them would depart too early for you. You'll need to see how those departure times align with the arrival time of the express boat and the departure time for the Gudvangen-Voss bus. The Fjord Cruise schedule is available here: https://www.norwaysbest.com/en/flam/things-to-do/fjord-cruise-naeroyfjord; the Lustrabaatane schedule is here: https://en.lustrabaatane.no/flam-gudvangen. (I don't think the Lustrabaatane ferries are really sold out.)

The bus schedule can be found here: https://www.skyss.no/en/.

The train between Bergen and Voss runs rather frequently, I believe, and in the past hasn't needed to be prebooked, because there were no assigned seats. Schedules here: https://www.vy.no/en

The express boat runs a few times a day but that is immaterial, as one would need the early one to be able to have any chance of doing this

It is certainly do-able, and there are more buses than the Skyss ones between Gudvangen and Voss as there are the long distance buses as well.
Just as with people who do the Norway in a Nutshell in a day It is a case of straight through connections with no real time to enjoy the places you are passing through, notably Flam. That is a shame, but if that's your style of travel or simply the time available that's fine.
Flam arrive 1340 from Bergen, Norway's best boat to Gudvangen leave 1500, arrive 1700, bus to Voss leave 1725, arrive 1845.
Trains to Bergen at 1914, 1951, 2011, 2107, 2206 and 2303. Except the 2011 and the 2207 they are commuter trains starting at Voss. The others are the trains from Oslo.
The train takes about 1 hour 20 minutes.
The connections are all robust, so it works, so in that sense no reason to advise against the plan.
For 6 people especially the "pinch point" which needs to be booked soonest is the Flam to Gudvangen boat as that is also one of the main boats in the Oslo to Bergen in a day itinerary. If that boat is full (and very likely to be) the whole itinerary falls over.
It could still work the other way round - train to Voss, for the 1010 bus to Gudvangen for the 1210 boat to Flam, then the afternoon Sognefjord boat to Bergen. Lots of commuter trains to get you to Voss by 1010.

Thanks. I’ll check out those times. As for Flam, I heard it can be a bit of a s-show, given the cruise ships and NIN traffic. So, I’m not necessarily real keen on spending any time in Flam anyway. Am I mistaken? In fact, I was thinking of trying to see if it was possible to skip Flam and instead use Aurland as our turn point.

I think it would be a crying shame not to go up to the head of the Fjord at Flam.

The bigger problem to me is that Aurland is a pre booked request stop only on the Gudvangen boat. We are all human, the master of the ferry could forget to call. That actually happened to a friend of mine on another ferry in that area, a genuine error. Luckily the harbourmaster could radio the ferry, after what were probably a few Norwegian expletives the ferry turned round. If that happened to you it's game over. Strictly it probably isn't as I'm fairly sure there is a late afternoon Aurland to Voss and Bergen bus but you get my drift.
In the middle of the day cruise passengers are mainly off either on excursions, or doing their own thing elsewhere.
Most nutshellers are only there for the time it takes to transfer from train to boat or vv.
But also there is the new Flam (Train station and pier etc) and the old village of Flam- what existed long before the railway and any tourists except the really intrepid. It's only 10 to 15 minutes walk to the Church (in old Flam) and it 's a totally different world.
Keep walking up the valley along the railway and you will be in real Norway, all but alone.
Another option is to get the Lustrabattane boat (the less touristy boat) at 1600 from Flam (more time to explore up the valley) arrive Gudvangen 1800, then the 1944 Vy Buss #450 to Voss (a long distance bus), for one of the late trains.
If that boat broke down then that same bus does call at Flam before Gudvangen, and there'll be the train as well.
And the boat from Bergen has gone half way into the Naeroyfjord on the way. So you'd have options, only you can decide if they are good options.
I'm probably more confident in getting myself out of a tight spot, and I'm one person not 6. For you I think it's best to change at Flam and stick with the Norway's best boat.

Just my 2 cents. We were in Norway May/June 2023.

Among the highlights of our 2+ weeks were the ferry through the Naeroyfjord from Gudvangen to Flåm and the spectacular views of the fjord from above during our minibus tour to Stegastein Viewpoint.

Go to a site like Cruise Mapper and plug in your dates to see what ships are in Flåm during your visit. Maybe you'll be lucky and there will be no mega ships.

BTW, your a group of six. You're already traveling with your own crowd.