If according to the Jehovah Witnesses that "there was a time when God was alone," who was with God at John 1:1? The source for my question is found at the Jehovah Witnesses own web site: The Watchtower :


  His Origin Was From Early Times: "Since all created things had a beginning, there was a time when God was alone."   


It then asks a rhetorical question: 


  Who was his first creation?  


They say:


  The last book of the Bible identifies Jesus as "the beginning of the creation by God." (Revelation 3:14) Jesus is "the firstborn of all creation." Source: Watchtower 2005 9/15 pp. 4-7: Who Is Jesus Christ?   


From the NASB it says at John 1:1, 


  "In the beginning was the Word, and the Word was with God, and the
  Word was God."


According to the context the "Word/Logos" is identified as Jesus Christ. John 1:1-14. 


  He was in the beginning with God and He was God.


If you're with somebody your a distinct person from that somebody. 

Moreover, there is only one beginning. John's "beginning" is referring to the Genesis 1:1 beginning. At the Genesis 1:1 beginning it states, "In the beginning God created. So the Genesis "beginning" is telling us WHAT HAPPENED "in the beginning." 

At John 1:1, "in the beginning" the emphasis in on WHO EXISTED "in the beginning." And verses like John 1:3, "All things came into being by Him, and apart (or without Him) nothing came into being that has come into being. Colossians 1:16, Hebrews 1:10 and even Revelation 3:14 backs up John 1:1. At Revelation 3:14 Jesus is the "arche/beginning" or the source and origin of the beginning of time and space. We get our English word "architect" from the Greek word, "arche."

The following confirms what was stated and if more information is needed, that can be suppied as well: John 1.
The answer is in The Watchtower that you cite in your question.

The Word” was with Jehovah God “in the beginning,” when “the heavens and the earth” were created. He was the one to whom God said: “Let us make man in our image.” (John 1:1; Genesis 1:1, 26) Jehovah’s firstborn Son was there at his Father’s side, actively working with him. At Proverbs 8:22-31, he is represented as saying: “I came to be beside [the Creator] as a master worker, and I came to be the one he was specially fond of day by day, I being glad before him all the time.”

Jehovah’s Witnesses believe that long before any angels were created, certainly long before and physical matter that makes up the universe  was formed, Jehovah created Jesus. If you need to say this happened outside of space and time to understand it that is fine. It was if space and time began with the creation of the universe.
We can better understand John’s words here if we analyze exactly what he said. Notice again that he wrote:

In the beginning was the Word.”

This, of course, was not talking about God’s beginning, for God had no beginning. (Psalm 90:1, 2)
It was the beginning of the things John was here discussing, including the creation of all other things by “the Word.”
Then John said: “The Word was with God.” A person who is with someone obviously is not the same as the one he is with.
So at John 1:1 when Jehovah began creating the heavens and the earth the Word (Jesus) was with Him.
Additional information on this closely  related question