Hello. It seems to me, that I found solution:
it is necessary to turn off the first iPhone and, then, turn on the second. If I understand correctly, when I turning on the second without turning off the first (which is on charging, for a example), there is a situation when two iPhones "saying", that they have the same phone number in iMessage. Of this, the second iPhone miss phone number for iMessage and => no text message forwarding. Should work also if the first iPhone will be in the Flight Mode – I will try.