"Because there was an accident, Tom went to the company late." I would say 事故があったので、トムは会社に遅れました。 As Ryoko-san said, 遅く行きました　and 遅れて行きました don't make sense in this context because the two sentences mean Tom intentionally went to the company late. But 遅れました means the accident made Tom go late.

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.