I play in a community brass band and have just taken up the trombone. I'd like to use my tenor trombone (in b flat) to play bass trombone parts where necessary. A lot of the bass trombone parts are in a different key. Eg: Bandology (arranged by Frank Wright) has the 2nd Bb Trombone in the (treble) key of C but the bass trombone is in the (bass) key of Bb. In Amparito Roca arranged by Aubrey Winter the 2nd Bb trombone is in the key of Eb but the bass trombone in the key of Db. I guess the music was written for trombone in the key of C? So transposition for a Bb tenor trombone to play Bass trombone is a full tone up? Am I correct?
If this is brass band music, with Trombones 1 and 2 in treble clef, bass in bass, I hope that ALL the bass trombone parts are in a different key!
Most brass band instruments are written in treble clef, as transposing instruments - yes, even the basses (tubas). Bass trombone is the exception, it gets bass clef at concert pitch.
If you're used to reading orchestra/wind band/commercial trombone parts in bass clef, read brass band bass trombone parts the same way. Written Bb is sounding Bb in first position.
Read brass band Trombone 1 and 2 parts a 9th down. Written middle C is Bb a 9th lower.
If you know tenor clef, there's a dodge. 'Brass band treble' is the same as 'Orchestra tenor' with two flats added to the key signature.
I think you are a little turned around by the terminology. Contemporary Tenor and bass trombones are pitched in Bb but normally are read in C. In other words, when you see a "C" written, you play a "C".
In brass band, there is some level of transposition still performed in some parts. If you see a bass clef part it should be in "C", which means you play a "C" when you see a "C". For the treble parts, it is probably in Bb, which means when you see a "C", you play a "Bb". Note that this is completely independent of the key of the piece.
The concert key of the piece is going to be what it sounds like, independent of what transposed instruments are part of the band. So a trumpet player might have a "C" written, but what sounds is a "Bb", a French horn player might have a "C" written as well, but it sounds like an "F", so on and so forth. The key signature for the trumpet player might be no flats/no sharps (Key of C) and the key signature for the hornist would be one flat (Key of F). However, for untransposed instruments, it would be "Bb", ergo, the concert pitch would be "Bb" even though the transposed instruments might be seeing a different key signature.
In this case, The Amparito is probably in the "concert key" of Eb. The 2nd trombone part sounds like it is written untransposed, ergo a written "Eb" sounds like an "Eb". The bass trombone part sounds like it is transposed for trombone in Bb, which makes everything down a whole step as you indicated. So a written "C" is going to be a "Bb" for that part.
In your first example, Bandology, it sounds like the 2nd trombone is in transposed "Bb" (so when you see a "C" you play a "Bb") and the bass trombone part is untransposed (when you see a "C" you play a "C").
If the 2nd Trombone part is in treble clef, you probably have a part written for trumpet players filling in on valve trombone. It's fairly common to see this in Baritone/Euphonium parts.
Tenor Trombone and Bass Trombone are both "C" instruments. That means the written pitch is the pitch that sounds. Compare that to a Bb trumpet, or an F Horn, or an Eb Alto Sax.
To directly answer your question, there is no transposition between tenor and bass trombones.