忘れた (forgot) is the past form of 忘れる (forget). 忘れちゃった is the colloquial way of saying 忘れてしまった (have forgotten). No need to say, FORGET is one of the tricky English words, which doesn’t necessarily accord with the time tense of the sentences. Don’t let it confuse you.

I would like to add that V(て form)+しまいます adds a feeling of regret to what you're saying. Usually, it's used in the past tense as what you are regretting has already been done. So, using your example: 宿題を忘れた。I forgot my homework. 宿題を忘れてしまいました。I forgot my homework (and I feel regret over it). しまいます is a verb as well, so it exists in plain form しまう and plain past form しまった. And, it is colloquially shortened to ちゃった.