Replacing the selenium rectifier in a vintage car battery charger
This is my first post here and I just stumble in your post, as I like to search about car battery chargers.
Answering  two questions and to suggest a way to improve your charger with minimal changes (at the end).

Q1: Can selenium rectifiers be replace by silicon rectifiers?

A1: Definitely YES, in fact, most repairs like that should replace. It will be an upgrade, making any charger work better.
BUT care should be taken to size properly the rectifier  selected. I believe the chosen KBPC2510 bridge is robust enough for your charger, as it is rated for 25A and the front plate of it says 10-12Amp.

Q2: Why the voltage seems to be higher than expected, as you measured and said:
“6V and 12V setting gave me 6.9V/11.9V open circuited and when connected to the battery the outputs rose from 6.8- 8V for 6V battery and 14-15.8V.” 

A2: Your older charger seems to be made essentially with a transformer and rectifier. It doesn’t have any filtering capacitor or regulation components. 
This case might be confirmed as you might have measured the “open circuit” voltage using a multimeter in DC voltage, when the average DC voltage is measured, which can be somewhat different than the real RMS value.
A rectified sinusoidal waveform with 11.9V of average value, has a PEAK value that is Sqrt(2) times this:
V_peak = 1.414 x V_avg = 1.414 x 11.9V = 16.8V
But a bridge rectifier has a voltage drop in two diodes (2 x 0.7V), so the maximum expected charging voltage is:

V_max12 = V_peak - 1.4V = 15.4V

If you repeat this for the 6V settings, you will find:

V_max6 = (6.9 x 1.414) - 1.4 = 8.3V

These small differences between what was calculated above and your measured values could be caused be [Average x RMS] and small variations in diode performance. 
So the battery behaves as large capacitor, and battery voltage is close to peak calculated voltages.

POTENTIAL PROBLEMS with this kind of charger - and how to alleviate them:


Does not have short circuit protection. This might become worse if older transformer is oversized and could deliver short-circuit currents bigger than 25-30A. This would result in overcurrent for modern bridge rectifiers, less forgiving to abuse (and spanking) than original selenium ones.
Does not have reverse polarity protection. This could also create severe overcurrent, damaging Si bridge rectifier.
Does not limit the current for deeply discharged batteries.
The final charging voltage might be too high, prone to battery gassing and shortening its life.


Possible contingency solutions, while 
preserving its original and antique simplicity &amp; robustness:


Install 12V headlamps in series - Installing 2 or 3 highbeam lamps in parallel among them (each about 50W) while in series with one of charger’s cabling, could limit the maximum charging current to less than 8A or 12A. 
This protects against short circuiting the terminals= lamps light up in 90% full brightness.
This also protects if terminals are connected with reverse polarity in the battery, the lamps would turn on even if charger is unplugged (turned-off) or lamps would light up “overly bright” if charger is already turned-on. 
This also limits final charging current and allow more time to turn-off charger when desired battery voltage is reached.
Install additional rectifier diodes in series - Measured 15.4V to 15.8V is somewhat too high to charge most 12V batteries. Final voltage should be about 14.4v for bulk charging.
One way to achieve this is to use the same 0.7V to 1.4V of voltage drop by the use of another bridge rectifier, selecting 


from negative to positive (2 diodes in series, working in parallel) to get 1.4V drop, or 
from shorted AC terminals to positive terminal (2 diodes in parallel) for 0.7V.



Note for Overheating management: 

Both rectifier bridges should be properly heatsinked, maybe even with forced convection (fan cooled.)
Heat generated by headlamps would benefit from forced convection too.

I hope these comments help you and other fellows.