遅れてすみませんって英語でなんて言うの？

The literal translation is, "Sorry for the delay." However, this is more formal, and not really what you would want to say to nursery school staff. "I'm sorry I'm late" is a better, more casual way to say this phrase. Usually, they are understanding, as long as it doesn't happen often.

>I am so sorry for coming so late.
*this is a very simple and sweat apology.

>Sorry for only coming now. I had some car trouble...
*By adding "I had some car trouble" is basically giving the reason why you came so late.

>My apologies for coming so late and for any inconvenienced caused.
apologies=a regretful acknowledgement of an offence or failure.
inconvenience=a cause of trouble or difficulty.

The important thing here is to make an apology and then also just add a little detail as to why you were late. Usually staff are not so interested in the reason you are late. Sometimes these things happen and cannot be avoided. Also nursery schools in the UK usually have in place a fine scale for parents picking up children late!
"So sorry, I'm late. My car broke down and I had to catch a bus!"
"That's OK, you're only ten minutes late."

Due to the inconvenience your lateness may cause to the staff looking after your child, it is always best to include an apology in this sentence. You could say 'sorry', 'I apologise', 'apologies', or 'excuse me'.

It is sometimes good to explain WHY you are late, but this is entirely down to you. There is no requirement on you to go into this kind of detail. For example:
"Sorry I'm late, I was stuck in traffic"

Instead of saying "I am sorry for the late " you say "I am sorry for been late."
Instead of saying "" sorry for my lateness" you say "Could you please excuse my lateness."

You could say something like

" I am sorry for not being punctual, I was running late".

"I am sorry the book is overdue."

" We arrived after hours, many hours later than we should."

The best way would be to start with an apology like above then just add the detail why you are like. Example
I'm sorry for being late, My car broke down or The bus was late.

"I'm so sorry I'm late.", can be used to express that you feel badly that you were not on time for an event. It can be used to tell someone that you know you were not on time and that you are very sorry that this happened. This can be used in everyday speech in informal situations, such as with friends or people you know well from informal environments.

"I apologize for being late.", can also be used to express that you understand that you did not arrive on time for something and that you are sorry for the inconvenience it caused another person or other people. By saying the word, "apologize", this statement becomes a bit more formal and can be used in slightly more formal situations, such as with co-workers.

"Please forgive me for being late.", is a statement that sincerely asks for forgiveness from the person or people who experienced an inconvenience as a result of your action. It expresses acknowledgment that you were wrong to allow this situation to happen and that you truly want to make sure that you receive forgiveness or a pardon from the other person or people who were affected. This can be used in both informal and formal situations, though it is not commonly used in everyday speech and can be considered a more formal apology.

"I'm sorry I'm late", "Sorry I am late" and "Sorry for coming late!"
These expressions are used when expressing your apologies (being sorry) for arriving somewhere late, or later than what you had agreed, very causally and widely used phrases for many contexts about being late.