Because I don't have time outside of work, the only chance I have of meeting someone is through work
仕事関係は「through work」で訳せます。
直訳すると「仕事を通して」。
したがって、「Because I don't have time outside of work, the only chance I have of meeting someone is through work」を提案しました。
出会いのきっかけは「chance of meeting someone」です。
婉曲的に、恋愛対象のイメージになります。
もし恋愛対象でなく、単に人に会うことでしたら「meeting people」に変えたほうがいいです。
I don’t really have time outside of work so I mostly hang out with coworkers.
I don’t have time outside of my job, so I usually hanging out with people from my company.
以外 - outside of, except
仕事以外 - outside of work, when I’m not working
時間はない - I have no time, no time
〜〜から - because 〜〜, therefore, due to
時間がないから - because I have no time, due to having no time
出会い - hang out, meeting
会社 - company
会社（の人）- coworkers
関係 - relation, connection, same
会社関係 - people from work, corworkers, people from my company
出会いのきっかけは会社関係が多い - most of my time is with coworkers, a lot of the time is with people from my company
仕事以外の時間がないから出会いのきっかけは会社関係が多い
- I don’t have any time outside of work, so I’m usually with people from my company
- I don’t really have time outside of work so I mostly hang out with coworkers.
- I don’t have time outside of my job, so I usually hanging out with people from my company.
イギリス 
