Lightning can be very private for senders from their merchants, and soon with coming improvements for receivers from the person paying them, but it is very weak against a truly powerful global adversary. This can be mitigated however. Payments stand out to a global adversary because that is the majority of traffic nodes will send, and the timing relationship from A to B to C to D, etc. These heuristics can be broken by nodes sending fake traffic to each other regularly. Fake traffic could take the form of a constant barrage of fake packets, simply replacing fake ones with real messages when payments are routed. This would make it impossible to correlate anything. Other options would be to add decoy messages that continue on after the completion of a payment, or opportunistically make payments when such decoy messages reach you.