Here's what to do:
Ethernet cable to the ZyXEL a PC (desk or lap -top)
browser to 192.168.1.1
logon 'admin' (default password is 1234)
WAN / Mode (set to 'Bridge')
LAN / DHCP (set to 'None')
Logout
Ethernet cable to the LinkSys (port 1 - 4) a PC (desk or lap -top). No cable to 'Internet' port
Start/Settings/Network Connections/Local Area Connection (double click)/Internet Protocol(TCP/IP)/Properties/Use the following.../IP address:192.168.1.2/Subnet mask:255.255.255.0/Default gateway:192.168.1.1/OK
Start/Run.../Open:cmd/"ipconfig /release"
browser to 192.168.1.1
logon admin (no password)
Internet Connection Type - PPPoE
User name: (value from BT Ireland)
Password: (value from BT Ireland)
N.B. You probably selected the above two values when applying to BT Ireland
DHCP Server: Enable
Starting IP Addresss: 2
Maximum number of DHCP Users: 5
Save settings
Cable ZyXEL to Linksys 'Internet' port
Power cycle (unplug/plug) Linksys
Start/Settings/Network Connections/Local Area Connection (double click)/Internet Protocol(TCP/IP)/Properties/Obtain IP address automatically/Obtain DNS server address automatically/OK
Start/Run.../Open:cmd/"ipconfig /renew"
HTH
Because the work laptop is operating correctly (it is remotely possible that it's last DHCP lease (unexpired) was from an identically numbered DHCP server, most which default to 192.168.1.1), the connection worked. Or, the registry key under HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\Dhcp is really foobar'd on the personal laptop, in which case -
WinsockXPfix might be yer man.