Sorry, I was very busy recently, so I forgot to reply to your message! How to say it in English?

If it is not particularly urgent and important, there is no need to use "apology" to mean "apology". Apologize is more formal, unless you have caused great losses to the other party, it is best not to use such a word, you can use sorry. For example, you can start like this:

Dear Mr.xxx/Miss,.

Dear Mr./Ms. xx,

How are you? I'm sorry to reply so late. I've been very busy these days and haven't checked my email for a long time. ...

Hello! I'm sorry to give you a reply so late. I've been very busy recently and haven't checked my email for a long time. . .

Personal opinion, for reference only.