for legal reasons I need to know what IP address was "allocated" at a specific date and time.
the BT router only has the technical data available since the last router reset. The router has been unplugged since the specific date. How can I find out the historic IP allocation?
all help gratefully received
In a nutshell, you can't.
I suspect BT may hold that information for legal purposes but it won't be available other than via a court order type route.
I doubt that BT will help you with this even if you are the account holder.
One way you may be able to get it is if you know the time and date you could obtain the IP address from an email that was sent from that computer about that time. Obviously you would need to have access to the sent email on order to do this but the IP address from the computer that sent the email should be included in the email header.
You could turn this on its head and look at the event logs for the devices on your network, e.g. if the device in question might be a PC or Mac, they may have this information in the OS event logs.
BT's home routers have a logging facility, but it's likely to be all in RAM so lost when the router is unplugged. BT's home routers use standard protocols (they are licensed and customized from other vendors).This information is you are looking for simply won't be conveyed to BT, ie BT's home hubs act as DHCP servers not DHCP relays.
All the devices on your home network are behind a network address translator. For IPv4 they'll share a single IP address. Even if you can associate a device with an address on your network, it's unlikely that anyone outside will know what the internal address was.
Since it sounds like a potential legal issue, you'd probably be best served by phoning BT during work hours.