I understand that Witnesses follow an unelected body of men called a "Governing body". However I'm unclear on whether or not they believe this governing body is in charge of the earth after their claims of Armageddon. Is this true?
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Often you won't get the same answer when asking random people from their congregations as some are more in touch with modern developments more than others, as you've just discovered.
Another thing is that the interpretations on core modern fulfilments often change: 2013 changed the governing body's role slightly. The "wheat and weeds" understanding also recently changed.
The answer is "No" they do not believe they will exclusively "rule the world", but are part of the 144,000 which serve as Kings and Priests under Christ, obeying him as co-rulers. They presently find themselves to to have a unique role among the anointed, that having changed to "Faithful and Discreet Slave", which dispenses food (teachings) in due season/proper time.
We do have an unelected Governing Body. Currently there are 8 men on the body. The term "governing body" is synonymous with the "faithful and discreet slave" (Matthew 24:45)
Jehovah's Witnesses believe that 144,000 men and women will be co-rulers with Jesus in heaven and rule over the earth (Revelation 20:4). Of course they won't be humans at that time. They will be spirit creatures like Jesus; a new creation (2Co 5:16, 17). The members of the Governing Body are part of this 144,000 and will rule in heaven if they finish their earthly life in a faithful state.