First of all, 行きました doesn't sound good to me in this case, I would use 家を出ました(left the house) or 会社に着きました(arrived at the office) because I don't understand what 遅く/遅れて is used for.
Since I'm not a professional teacher, I'm not very sure which is correct, however, we use both way in everyday conversation. For example, いつもより30分遅く家を出ました/ いつもより30分遅く会社に着きました/ いつもより30分遅れて会社に着きました, all of them sound natural to me but I wouldn't say 事故があったのでいつもより30分遅れて家を出ました because I think the accident didn't affect for Tom to leave his house. If He had problem in your house before he left, for example, he lost his key, I think we can use 遅れて like トムは鍵を探していたのでいつもより30分遅れて家を出ました.
遅くand 遅れてare, as you know, adverbs. 遅く comes from 遅い(adjective) and 遅れて is from 遅れる(verb). I think there isn't any big differences in everyday conversation.
I hope some professional teachers answerd to you.