I'm beginning to wonder whether it is incompatible versions of software that is causing your problem. In November last year I changed the port used by the Apache web server on the BridgePal PC from 80 to 1357. This was to avoid clashes with other software that uses port 80. The change was made in BridgePalServices release 1.6 and BridgePal Server v1.4.0 . At the same time I released v1.4 of the BridgePal app, which was a necessary upgrade for compatibility with these releases.
I note Lingwood went live earlier than this so certainly it would have originally been using an earlier version. I also note that you say the ZTE Blade is running BridgePal 1.3.1, so if that works with the Lingwood laptop it would suggest that the laptop has not been upgraded and therefore runs Apache on port 80.
The new laptop presumably has the latest version of BridgePalServices installed, with Apache running on port 1357. That would explain why phones which work with one PC do not work with the other, even though it has the same ip address. You would need to upgrade the software on both PCs, and all phones, to the latest releases.
Incidentally, the bridgepalBroadcast configuration mechanism is not supported on BridgePal app releases prior to v1.4
If you can establish whether my above assumptions are correct I can advise you on the upgrade procedure. You can check which version of BridgePalServices is installed by going to the control panel. Look for "Programs and Features" and find the entry for BridgePalServices, which will show the version and the date it was installed.
John