I had to leave the celebration part-way through because I didn't think I was going to make it on time to pick up my son.
ーI had to leave the celebration part-way through because I didn't think I was going to make it on time to pick up my son.
「息子のお迎えに間に合わないと思ったので、祝賀会を途中で抜けてきた」
to leave ... part-way through で「…を途中で抜ける」と言えます。
to make it on time で「時間に間に合わない」
ーI had to skip out of the party before it was over, otherwise I might not have been on time to pick up my son.
「食事会が終わる前に抜けなければいけなかった。さもないと息子を時間通りに迎えに来れなかったかもしれない」
to skip out of ... で「…を抜ける」
to be on time to ... で「時間通りに…する」
ご参考まで！
I ended up leaving halfway because I didn't think I would be able to make it on time.
I ended up leaving halfway because I didn't think I would be able to make it on time to pick up my child.
ご質問ありがとうございます。
「間に合わなさそうだから途中で抜けてきた」はそのまま英語で「I ended up leaving halfway because I didn't think I would be able to make it on time.」と言います。
また、「子どものお迎えに行った」と追加したいなら、そうすると、「I ended up leaving halfway because I didn't think I would be able to make it on time to pick up my child.」になります。
ご参考になれば幸いです。
カナダ 
アメリカ合衆国 
