Hi frosty,
You should reduce R2 and R3 at least by a factor four.
A good value for R2 is 220
hm: to 240
hm: , then calculate R3.
There shouldn't be any problem if the battery ( or a load ) is
permanently connected, but when no load is present the output
voltage will rise well above the calculated value because you forgot
that the LM317 also needs current to function and there isn't a
direct path to ground. They usually solve this problem by taking
lower resistance values for the voltage divider. You can also solve
this problem by connecting a small permanent load like a led and
it's series resistor. 5 mA through the led is enough to solve this
problem.
You also forgot that each battery charger needs to have a reverse
blocking diode in case the mains runs out of power when you're
charging a battery.
The 2200 µF capacitor will charge up to the peak value.
on1aag.