All of the BA Lounges in T5 share the same internet pipe (not sure if that includes the T3 BA Lounges, I think it does, but I can find out for sure), so no lounges or locations have more bandwidth available than any others.
All the AP's are now quite old, and not really up to the amount of devices that connect to them in busy areas - back when it was spec'ed, 1 device per user was the norm, now people are connecting 2 or 3 devices each which means more devices having to share radio time slices, to a point that they become unusable

And before you go and blame BA, all the AP's in the terminals are BAA's, including the lounges. But I would still complain, as it might actually get upgraded one day.