One of the skills every organist needs to develop is the ability to transpose since improvisation and transposition are so closely related and shed so much light, one upon the other [See blog, Improvisation, Parts I, II, III, IV, V].
Transposition is also a skill that must be practiced in order to be mastered, just like improvisation.
There are 3 ways that an organist can use to learn to transpose: by ear, by clef, or by analysis (interval).
Some experience with all 3 can be very useful to improvisors.
Your ear is a great asset in transposition and is your best friend; it will lead your brain if and when your brain starts working too much.
If you've learned a melody by ear, then it may be easy to transpose it by ear into a different key as you're improvising.
Harmonies, especially complex ones, can be much harder to transpose by ear as you're improvising and can be the slowest way to practice your transposition, but your ear will always be how you judge if your transposition is correct.
A helpful and inexpensive resource which teaches you how to think as you modulate from one key to another is Max Reger's little book called "Modulation."
It's an easy read which has very short exercises of 2-4 bars which shows an improvisor how to move from one key to another quickly and easily [See Improvisation, Part VI].
These little exercises, which start out in a certain tonic key and move to some other key, can be transposed mentally starting from any other tonic according to formula which is given with the exercises.
When using clefs to transpose as you sight read, the simplest transpositions are those by a half step; it's relatively easy to read a hymn or other piece written in Ab Major in A Major by simply changing the key signature, mentally.
Likewise, moving from E Major to Eb Major requires only a change in key signature, as the noteheads on the page are exactly the same; all we have to do is pay some attention to the alterations.
We can change the clef mentally and read the music in a key further away, but, sadly, a goodly number of musicians today are only fluent in reading treble and bass clef.
Unless we read from a lot of early music scores, we don't typically read from the other C and F clefs; there are enough different clefs that any line on the staff can actually represent the same pitch [See blog, Open Score, Parts I, II].
One of the best references that teaches how to read these clefs is "Preparatory Exercises In Score Reading" by Howard Ferguson, et. al.
Being comfortable with reading the different clefs makes it much easier to transpose pieces into more distant keys as we sight read.
Learning to read by clef reinforces reading by analysis, or interval.
One form of transposition would be to consider the interval that each voice moves; this can be very helpful when transposing a single melody or theme as we sight read but also for complex harmonic structures.
Recognizing that a voice moves a half step down (photo) might be easier to see than reading the part in a new clef which shows a half step movement.
In tonal music where we can analyze harmonic function, knowing that the original is in a certain simple progression usually makes it easier to play the proper notes and progression in a new key.
This is where Reger's little book is helpful to transposition.
The further away on the Circle that we have to transpose a piece, the more likely we are to rely, in addition to using a clef and our ear, upon some form of analysis by interval as we sight read it.