Rather than "branch" I would say routs. Aikido's spiritualism is in its routs.
From the little I saw here and there, it is the Sensei who would emphasize spirituality, not he organization.
As well as I know, the amount of spiritual teaching is not something that is decided by any particular aikido organization.
No not really - the question is completely fair. I would say that all organizations contain some spirituality in that there is bowing to the kamiza but beyond that there is a huge variation. It is probably best seen not on how much is discussed in the class but how their students rate the importance. If we had a straw poll asking what organization are you from and how high do you rate spirituality the answer wold become clear pretty quick and putting myself out on a predictive limb I would say that the pre-war styles would be less inclined to identify themselves as spiritual.
To start with Shodokan Aikido - spirituality low.
That said my teacher for a time taught at Ueshiba's old Omoto-kyo dojo and the Kamiza at Shodokan honbu is blessed by Omoto-kyo priests once a year. Just that Aikido practice is primarily technical and any lectures reflect that.