I'll let you know if I remember.
思い出したら言うよ。
I'll let you know when I remember.
思い出した時に言うよ。
let you know が「教える」「伝える」ですね。
if を使うと「もし思い出したら」、when を使うと「思い出した時に」の違いがありますが、ほとんど同じニュアンスです。
例：
Kevin: What did she say was going to be on the test?
テスト範囲どこだって言ってた？
Erik: Ah, I was listening when she mentioned that... I'll let you know if I remember.
あ〜。それ言ってた時聞いてたよ俺…思い出したら言うわ。
Kevin: Wow, that's a great help.
使えねーな！
As soon as my thought comes back to me, I'll let you know.
"As soon as my thought comes back to me, I'll let you know. " - In this sentence you are implying that you have lost your train of thought but you will let the teacher know as soon as you remember.
Example sentence:
Teacher: Yes ~, what did you want to say?
Student: As soon as my thought comes back to me, I'll let you know.
"As soon as my thought comes back to me, I'll let you know. " -「考えていたことを忘れてしまいました、思い出したら言います」という意味合いです。
例:
先生: Yes ~, what did you want to say?（はい ～、何でしょう）
生徒: As soon as my thought comes back to me, I'll let you know.
I cannot remember but I will let you know when I do.
I don't recall but I will tell you if it comes back to me.
There are various ways you can say this.
Recall and remember are near enough the same thing but recall is possibly a more advanced way of saying it and it will definitely impress your English teacher!
'Comes back to me' is another way of saying 'When i remember'.
これはいろいろな言い方ができます。
'recall' と 'remember' はほぼ同じ意味ですが、'recall' のほうが上級者的な言い方かもしれません。
使ったらきっと先生も感心しますよ。
'comes back to me' は 'when I remember' の別の言い方です。
My mind has gone blank, I've forgotten what I was going to say
It is common to say "it's gone" when we forget what we were about to say, meaning it's gone from my mind. Then you would say "I will let you know when it comes back
It is also common to say "my minds gone blank". We can say this when we forget something we are going to say or if we forget when we are making a speech or giving a presentation.
E.g. Excuse me, my mind has gone blank. Just give me a minute and I will remember what I was going to say.
"it's gone"はよくある表現で、何か言おうとしたことを言えなかったときに使います。そのあとに、"I will let you know when it comes back : また思い出したときに言うよ、と繋げることができます。
"my minds gone blankというのはかなり一般的な表現です。プレゼンやスピーチの際にも何かいいかけたときに忘れてしまったときに使うことができます。
E.g. Excuse me, my mind has gone blank. Just give me a minute and I will remember what I was going to say.
例文翻訳割愛
We use the word "slip"when we forget something. When you can't remember something at all, you can also use:
"It totally/completely slipped my mind!"
"slip" means "fall", so it fell out of my head and I forgot
何かを忘れた時には"slip"という単語を使います。何かを全く思い出せないときには以下も使います:
"It totally/completely slipped my mind!"（完全に忘れてた！）
"slip"は"fall"（落ちる）という意味ですので、頭から落ちて忘れたということになります。
I forgot what I was going to say. I'll tell you when I remember!
This is very irritating when it happens, but it's something that may happen to anyone - especially as you get older! Forgetfulness is sometimes referred to jokingly as '" senior moment."
"Oh dear, I think I just had a senior moment!"
"Why, what happened?"
Well, I forgot what I was going to say. I'll tell you when I remember!
"OK!"
これもし起こったらすごくイライラするよね。けど、年を取れば誰にでも起こりうるだよね。 わすれっぽさというのは、ジョークも込めて、senior moment と呼ばれることがあります。
"Oh dear, I think I just had a senior moment!" あれ、ちょっとボケちゃったかな。
"Why, what happened?" え、どうしたの？
Well, I forgot what I was going to say. I'll tell you when I remember! 今言おうとしたこと忘れちゃった。思い出したら言うよ。
"OK!" おっけー
I can't remember right now, but as soon as I do. I'll let you know.
I'll let you know once it comes back to mind.
1. I can't remember right now, but as soon as I do, I'll let you know.
This statement is telling the teacher that as soon as you can recall the information you will share it with him/her.
2. I'll let you know once it comes back to mind.
The phrase "comes back to mind" means to remember something. so this statement means that you will tell the teacher once you remember.
1. I can't remember right now, but as soon as I do, I'll let you know.
(今すぐには思い出せないですが、できるだけすぐにあなたにお知らせします。)
この言い方は、情報を思い出したらすぐに、情報を共有することを先生に伝えています。
2. I'll let you know once it comes back to mind.
(すぐに思い出したら、お知らせします。)
「come back to mind(心に戻ってくる)」というフレーズは、何かを思い出すことを意味します。
この言い方は、思い出したら先生に教える事を意味しています。
I am sorry, I don't recall but I will tell you if it comes back to me
The second sentence I would consider as the best option in your situation because it's super polite and you promise to tell the person if you recall. Although, the first one could be used too.
アメリカ合衆国 
南アフリカ 
イギリス 
カナダ 
