We worry about two things for this type of bathroom: A) Will the joists hold everything up? and B) Will the joists deflect (bend) too much and cause the floor or ceiling to crack?
Obviously, the critical support is located under the tub/shower. However, there are other areas of concern too: 1) size and grade of joists, 2) bearing wall location opposite exterior wall, 3) finish floor material and finish ceiling material below.
1) The Code requires a residential floor to support a “live load” (people) of 40 lbs. per square foot plus all dead loads. The floor dead load (permanent material) seems to be about (from the top down): ceramic tile floor: 6 psf, underlayment: 2 psf, subfloor: 2.5 psf, joists (assuming 2x8 @ 16” oc: 1.5 psf, 6” batt insulation in floor: 1.5 psf, ceiling finish below (gypsum board): 2.5, plus misc. (lights, ducts, etc.): 1 psf, for a total of 17 psf. If you add that to 40 psf, you get 57 psf (plus the weight of the water).
2x6 (old sizes) joists @ 16” oc spanning 11’ (average of your guess) will support about 51 psf and 2x8’s @ 16” oc will support about 93 psf spanning 11’
As you can see, 2x6’s are close and 2x8’s are fine for the entire room, except for the added water (tub/shower area).
If the 60 gallons is correct, it would add about 500 lbs., not 585 lbs. However, adding 585 (tub water) plus 200 lbs. (shower water) plus 350 lbs. of people weight, you get about 1,135 lbs. of live load. For the 3’ x 8’-6” space of the tub/shower, that works out to about 44 psf. (Close to the 40 psf required by Code.)
Because the tub/shower area exceeds the allowable loads supported by 2x6 @ 16” oc, it will deflect and crack the tile floor and gypsum board ceiling below. I recommend adding 1 joist on each side of the existing joist under the tub. Make sure it fits tight against the subfloor, because new 2x6’s are smaller than your 1930 2x6’s.
If the span is 12’ and the floor is 2x6’s, then you’ll need to add 1 joist between each of the existing joists in the whole bathroom too. (2x8’s @ 16” oc spanning 12’ supports about 78 psf, which is fine.)
Oh, one more thing. There’s a lot of load on that bearing wall. I’d make sure there’s at least 1 stud directly under each end of those joists under the tub/shower area and you have adequate footing under each bearing wall too.