IME some queens move down, some don't.  Brood chamber reversal is not
something to do just automatically.  It can cause problems for the bees,
especially if cold weather moves in after the manipulation.  With the brood
nest split the bees can have a difficult time maintaining temperature
within the nest, and brood can be lost to the cold.
 
I have had the misfortune of finding this out the hard way, once, when I
"reversed" a whole yard of colonies on a balmy late April day.  A cold
front moved in that night, and the low temperatures persisted for several
days.  Next time I checked those colonies, most were basically starting
over again, having had a large portion of the brood die.
 
Now I wait 'til the beginning of May, and make sure the weather is settled.
Some queens will by then have a nice lot of brood in the lower as well as
the upper brood box.  Still, by and large, I reverse them at this point.
The reason is, even if the colony has moved down and is occupying the lower
brood chamber, the lower and outside edges of the combs are still largely
unused and may still hold a lot of garbage/mold that accumulated down there
during the winter.
 
Raising the lower brood-box up above allows me to remove old burr- or
ladder-comb, and gets the bees to repair and clean up all of the combs.
Then the space is better utilized, I find.  This is a natural result of
their tendency to work upwards anyway.
 
Without reversing, the bees might not make full use of the comb area at the
bottom, and the combs would end up worn, chewed and filthy from just being
used as a passageway.  I have seen swarms from colonies which never
actually moved down to utilize the entire comb space at the bottom.
 
On the other hand, if the brood nest is spread between the two chambers,
and the cluster is not very large, reversing is not a good idea, yet,
anyway.  In that case you end up just splitting the brood nest with a big
gap between the parts, and the colony cannot deal with this well.
 
So... not too early, and according to each colony's condition -- that's my
take on it, anyway.
 
JG              Freeville, NY