I'm so glad both your working life and private life is going well.
I'm really pleased that everything is going well for you both at work and in your private life.
「公私共に全て順調でうまくいってるみたいで安心した！」＝ I'm so glad both your working life and private life is going well. / I'm really pleased that everything is going well for you both at work and in your private life.
ボキャブラリー
公私 ＝ working life and private life / business and personal life
うまくいってる ＝ going well
安心した ＝ glad (for you), pleased (for you)
全て ＝ everything
"It's a relief to see that things are going well for you both personally and professionally!"
「公私共に全て順調で、うまくいっているみたいで安心した」という表現は、英語で次のように言えます：
- "It's a relief to see that things are going well for you both personally and professionally!"
「公私ともに全てが順調で、うまくいっているのを見て安心した！」
イギリス 
アメリカ合衆国 
