I'm confused.
The picture you showed has the transformer labelled like this:
30 VA
Primary: 0-115 V (Blue, Grey), 0-115 V (Violet, Brown)
Secondary: 0-6 V (Orange, Yellow), 0-6 V (Black, Red)
Can we confirm that the secondary windings on your transformers are marked 0-12 V, 0-12 V, rather than 0-6 V? And the transformers are still 30 VA?
If so, for each transformer individually, the max current in each winding would be 15 VA / 12 V = 1.25 A. Combining transformers and putting all secondary windings in parallel, that would give a total of 1.25 A x 4 = 5 A.
You might get a bit more current by halving the primary voltage on each transformer, but resistive losses go with the square of the current, so trying to operate at 30 VA with halved voltages will give you twice the current and four times the resistive heat generation. So 10 A at 6 V AC might be pushing it.
Now you say you want to charge batteries, which presumably means rectifying to DC? If so, you have to considerably de-rate the transformer. I would think 10 A DC is highly unlikely, certainly not at a reasonable voltage. You are probably better being conservative and assuming 5 A rectified DC, and likely not at 6 V. Maybe at 5 V DC perhaps, but that would need some measurements and design calculations.
For DC applications, transformers need to be uprated, with both higher VA rating and higher secondary voltage.