""By the time I'm going home, the rain shall/will have stopped." Are both shall/will allowed in current-day BrE, in this context, in all registers?"
In that sentence, I think it should be "will".
The rules I learnt at school were:
Normal: I shall, you will, he will.
Emphatic: I will, you shall, he shall.
But I think nowadays the difference with the first-person is being eroded in colloquial speech.
Cheers
Tony
Tony Mountifield