Hopefully i can help..
If you havent done the 3 minute rule disconnection and reconnection then it possible you need to run through the bsi reset which i highlight below..
Peugeot are fairly sensitive to battery changes/jump starting - if the correct procedure isnt followed then you can run the risk of corrupting the bsi software or loosing key coding where the vehicle immobiliser simply wont shut off hence the engine wont start... Try the 3min rule again and then follow the reconnection procedure below..failing that the only way forward from that is to get it diagnosed to exactly whats gone wrong the likely hood the immobiliser isnt picking up the keys or a problem has surfaced in the bsi due to the battery fault.power surge or spike..
The following procedure is the battery connection unless told otherwise by Peugeot or Product service. Failure to follow this procedure may result in damage to the BSI’s embedded software during reconnection.
BSI start up
So as above give that a try - failing that the only alternative is the diagnostic route to see whats happened if the BSI has been damaged or lost key sync which can be quite common..I do hope this helps