Replacement Transformer for Vintage 6V/12V Battery Charger, Dual Voltage Primary, 3A Output

Question on what type of transformer is used in a battery charger. The transformer that I need to replace looks weird. The non polarized ac plug has one wire soldered to a post on the transformer and the other ac wire goes to a switch for 6v or 12v. The other two sides of the switch has two small wire from the transformer. The output of the transformer one wire that goes thru a meter and out to the battery. Then there are two bare wires out of the transformer that go to the case, thru a diode and out to the battery. Older charger. What type of transformer do I need to buy? 3 amp output. It seems like it is a dual voltage primary. So it seems like a 220v/110 primary with 12v out on secondary. So when the switch is set to 6v it would like putting 110v on the 220v tap and only getting half out. Any help on where to get one and the type would be nice. If you need a picture let me know.

I am here assuming that you have sufficient knowledge to safely carry out basic tests on mains voltage equipment, but if you have any doubts about you own knowledge and qualification, get the unit tested by an electrician or electronic technician.
Never have the unit plugged in while the case is open (unless you are already a qualified electrician).
If you can post a couple of clear close-up pictures, it may help us to confirm your circuit description.

A few questions to get a clearer picture:
How do you know the transformer itself is faulty ?
Have you checked that the diode and the switch and the ammeter, and the connections between them and the transformer, are OK ?
Have you checked the fuse, if the unit (or it's plug) has one ?
What is your mains supply voltage, and what is the input voltage marked on the unit ?

Check the transformer by unplugging the unit, and connect a multimeter set to ohms across the mains input:
what does it read when the unit is switched between 6V and 12V ? If there are two readings, one about twice the other, the primary winding is OK. (Assuming any input fuse is OK, and that your circuit analysis is correct.)

With the unit unplugged and opened up, what is the meter reading across the secondary winding ?

If the exact transformer type is difficult to find, you may consider replacing it with a standard dual output 6V and 12V transformer, and modifying the circuit as per the attached standard circuit for such units. For this circuit, the type of transformer may be described as "12V centre-tapped" or "6-0-6V" or "6+6V", and for a 3A, 12V output, its power rating must be (at least) 36VA.
(Go higher with the power rating if you want your battery charged faster, but obviously within its limits. Car battery chargers are commonly up to 15A.)
Diodes are very cheap, just make sure their current and voltage rating is equal to, or above, what you are using them for, so have at least 25V diodes (as the peak voltage of the transformer output is about 1.5 times its nominal value).

Oh, and by the way, check whether you can buy a new charger unit for less than the cost of a replacement transformer !

Bonus: Diodes: This way round.

Took electronics in high school. Got my E-2 license 10 years ago and work with industrial controls. Now I think the transformer is a 220v/110v primary with 24v output. Because the 6v 12v switch is on the primary side. So if you had the switch set for 12v the 110v tap, you would get 12v on the secondary. Now if the switch was on 6v the 220v tap would be used (with 110v applied) and 6v would come out of the secondary.

To battery charger transformer, actually to say, I have no knowledge about this tool. But I am familiar with bestek 300w power inverter. This tool is my camera battery charger. The easy way for me to use this tool is to connect with car cigarette lighter charger outlet. Then I can charge camera battery in car. Hope this point is useful to you. Have a nice day.