The character of any tube can vary depending on the design of the amp/preamp. I have heard preamps using 12AU7s which sounded downright chilly, neutral, and warmish. Much depends on the manufacturer and its type of tube (often the same brand has multiple models and dates of production which will sound different).
I'm guessing you are aware that the 12au7 and the 6sn7 are of similar design except for pin output, but the perception seems to be that the 'warmth' attributed to the 6sn7 will follow to the 12au7. This may not be so. A good reason to use an amp with the 12au7 is that the availability of that tube both as a NOS or new production as well as cost. No so much for the 6sn7 - NOS are hard to get, low noise ones especially and they are expensive. There are new production 6sn7 now which are reputed to be quite good (Tung sols for example) but I haven't heard one yet. If one has an amp/preamp which uses this tube I expect one could find a tube which would err on the warmer side, but not so much as a preamp designed around a 6sn7.