Hi, everybody.
I am also a learner.
I'm going to give my two cents worth.
If I make any mistake, please feel free to criticze me.
Tomasz Klimkiewicz is right the word "until" is used wrongly.
"he would help me until he came back"
That means he would be available and help you from now to the time when he came back. After he came back, he would not help you. I'm afraid it will never be correct except for jokes.
His alternative is correct. Another one you may use is "when".
Back to your question, it is a question about reported speech.
Christopher promised he would help me until he......back
a)comes
b)came
c)will come
d)would come
Since the sentence uses past tense, so answer (a) and (c) must wrong.
This leaves only (b) and (d).
You may think, since the case of "his return" has not realised yet, so we should use "would come". It is true but partially. The answer should be (b) came.
There is tense simplification in subordinate clauses. Subordinat clauses refer to the clauses which is not the main/parent part of the statement.
So if the main verb of a sentence make sit clear what knid of time the speaker is talking about, it is not necessary for the same time to be indicated again in the subordinate clauses. It is also used to make the sentence neater and less clumsy.