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I've seen more than a few comments about having to learn "new fingerings" when taking on a new horn in a different key, Like a BBb player buying their first F.

A typical pro response is, learning new fingerings is not that hard. And eventually, just picking up the Sousa and the BBb side of the brain kicks in, and picking up the concert horn, the CC side does the job. Maybe, this is similar for a Bb treble clef player (from trumpet or brass band lit) learning bass clef, or the other way around. So it seems to me, that novice just has a bit of fear of the unknown, or that learning something new will make me unlearn something I now know. And I think most of us would agree that no one is actually learning "new fingerings" for F, CC, Eb, BBb or switching from treble to bass clef, no more than if you switch from BBb sousa to Bb euphonium because the fundamental is always open, and first valve is a whole step, etc.

Most sax players would probably say (because their music transposes to keep the same fingerings), that the hard part of playing different keyed horns is the different blow of a smaller or larger mouthpiece and horn and the expectations for a tenor, bari, soprano, alto sound, not what pitch comes out when you press whatever button.
As a BBb player, when I was suddenly put on the spot to play an Eb backfire tuba for a Civil War band on period instruments, I used kind of a transposed treble clef shift (reading 3rd space written bass clef Eb as a Bb treble clef 3rd space C and change the key signature, which then sounds Eb on an Eb horn). Kind of like using tenor clef to read treble clef, for vice versa. Now that sounds exhausting, and it was a taxing mental leap, but temporary, and it got me through a rehearsal. After a short while, I didn't need the crutch anymore. And I never sat there and stared at a fingering chart for Eb.

So for those who have crossed this border, and lived to tell about it, what was your transition like (from BBb to CC or CC to F etc.), and do you have any "tricks" to share that got you over the initial hump? And what is going on in your head or playing when, like in orchestra or quintet, you switch among 2 or 3 keyed horns in a performance?

Sounds like you have a good approach very like my experience; just stick with it and you’ll improve, I second weaning yourself off transposition tricks as quickly as you can (a gig with a wire choir in sharp heavy keys will drive the point home!). Regarding switching horns on a gig, I tried it for a while (oh man I spent $$$ and want to show off this shiny new toy....) but eventually decided to avoid switching on the fly unless absolutely necessary. The reasons being: for me at least, the gig is always more of a brain fart generator than the practice room; big horns get cold and go flat when sitting (or sharp if in the sun!); finally schlepping several big horns around and managing them on stage is a pain.

Like most people, I started on BBb. I wrote the fingerings in when I made the switch to CC just before college, and my teacher told me to approach it as a different instrument altogether. A lot of learning the new key was solfege. C and G fingerings were never written, and my brain clicked into C in no time. Learning F took even less time because the fingerings felt like a hybrid between BBb and CC.

Pros of seeing different keyed tubas as different instruments: it made the switch to a 6/4 easy. Seeing it as yet another different instrument, getting the right air and accommodating other BAT tendencies was a breeze.

Cons of this method: tubas that look like each other hurt the brain for a bit. My junior year of college I got to use the school's PT-4T. As I would pick it up and bring it to my face to play, my brain said "PT-10 with a slight low C problem" instead of "5/4 rotary CC tuba" because they looked so dang similar when coming up. That led to a lot of fracked first notes.

I'm not a pro on brass but I "just do it." Last set of new fingerings was learned at 68. Ya can, ya know? You plenty smart enuf.

Graduating from the piano my brass life started with the trombone. Playing euphonium was easy, just calculating the valve combinations out of the familiar slide positions (bass clef). My start on tuba I made on a loaned 5-valve C. So I had the challenge of some new valve combinations (and 4+1 instead of 3+1) and reading "one tone aside". After some months I bought my own 3+1 compensated Eb tuba (ah, just like euph, just start at a different point!). Later I had my first euph-student which was reading treble clef...
For me it worked just to know how the instrument works and which note is where to find. And I read music not fixed to "note=fingering" but "the last note had fingering X so the next note in interval Z should be on fingering Y". This means, I play by reading the music as a singer and fingering through by ear. So it is possible for me to survive playing Eb tuba, Bb Saxhorn with 3+1 and 3+2 valves, 3+3 french c-tuba, 3+1 c-Saxhorn and Bb-ophicleide when needed on the same day. With this technique it is also simple to transpose into any key, just start with a different note and you can still play what you see. You only need to know the chromatic scale on your horn (and leave the notes you don't need out)...

For me what worked when going from BBb to CC was digging out the beginner method books and just start playing through them. It didn't take long for the fingerings to set in. It would throw me a fit if I had to switch between different keyed Tubas on a gig. It usually takes me going through a few scales when switching horns at home to get the new key in my mind and under my fingers.

People talk about "switching", and I noticed that F, E, Eb, D, Db, C, and B are the same on BBb and F, so I didn't think of it as much of a switch.

If you have learned to read "Bb" music on a BBb tuba, the CC tuba is not so much of a switch either. If you can already play an F tuba, the shared fingering come into play there as well.

The brain cells required to play Eb have all been spoken for by other activities.