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The Arban written for Trombone was written for a Bb instrument, and while you are correct, tubas do not transpose, the Bb book was written with Bb instruments in mind. The newer Tuba version has all of the same basic material (except for the back of the book with slight differences) just that it is written for an instrument written in the key of C. In my opinion it does not make that much of a difference for most of the book, however there are certain Technique studies that work better when written in the correct fundamental pitch of the instrument. ..... e.g. If I were to play any given exercise on my C tuba in the tuba book, they would be the same fingerings as a Bb tuba would play in the trombone book. I hope this makes some sense...

When practicing etudes such as those in the Arban Method, transposing them to different keys is a great idea anyway... so it doesn’t really matter which book you have. Using the trumpet book, you can learn to read treble clef (down two octaves), which can be useful.

More importantly, the standard trombone version of the Arban Method is missing many exercises.

I don’t own a copy of the Tuba version, but a few of my students have it and it looks good to me. I would recommend this version.

Having just started studying Blazhevich (American notation, starting in C major) on BBb, I can appreciate the different CC and BBb versions of Arban.
99% of methods start with "easy" fingerings, and work towards more difficult ones. A version starting in C major, when played on a BBb instrument will have more awkward fingerings and, at least on my instrument, more tuning issues from page one. Better to start in Bb, and depart from there. Fingering-wise, you're then following the plan Arban devised.

I also started with Uber, and, being also for bassbone, it starts in Bb. I find it easier to play despite the much larger compass.

An added benefit of playing out of a book not intended for your instrument is octave transpositions. Playing out of the trombone book transposed down an octave on a BBb tuba helps you hone a skill that any person who plays church gigs and the like. Often on these calls there won't necessarily be a part for tuba but rather a trombone two part, reading the bass line off of a piece for choir, reading piano music, etc. The more fluently you can do this, the better you will perform at your gig (especially when there are limited rehearsals), and the better your perform on your gig, the more likely you will be called for another one!

It's for most of us to take this skill for granted, but it's likely that there are few of us who couldn't be better at it.

If you use the cornet/trumpet version, learn to read TC and Bb trumpet fingerings, you can play it on whatever instrument you have. Fom piccolo trumpet to contrabass tuba, it will always lie in the tessitura of the instrument you're holding.

You'll have another couple of benefits--the availability of the huge repertoire of trumpet and horn music and the ability to play brass band parts.

You can also take what you've saved by buying the trumpet version and buy yourself a few beers as consolation.

I study with Dr. Young @ UW - Eau Claire, so needless to say I have the book. I played that book for 4 years with a BBb, and just recently ( about a month ago ) switched to CC. It's really not that big of a difference. Some things that were hard on BBb are now easier, and vice versa. It's a great book and I suggest getting the tuba version. There are things in there that have helped my playing a lot. Dr. Young knows more about the tuba than a LOT of people and he put tons of time and effort into that book. He knows what he's doing, and the book shows it.

I'm with ChuckG here- the trumpet version allows for any key of tuba. The Roger Bobo "Mastering the Tuba" volumes do this also, it's great.

Although for a student who hasn't mastered reading bass clef, I'd go for the trombone version (assuming he's on a BBb horn).

The original trumpet version is cheapest and contains all the stuff.

If you're going to use a trombone-euphonium version, how about the new edition by Brian Bowman and Joseph Alessi? It's written for Bb bass clef instruments, and unlike the old Mantia-Randall edition, it does contain every bit from the original trumpet version--the duets, characteristic studies, and solos. I use it daily.

Here is my $0.02:

I would use a beginner trumpet book to get used to the new fingerings, and THEN start to apply that knowledge to the Arban's, at some future date.