In play I have a Skatewing Spy and a Skitter Eel which is enchanted by Sky Tether. I activate adapt, which puts a counter on Skitter Eel.

Does this change the time stamp of Skatewing Spys static ability, giving the Eel flying regardless? Or will the Eel only fly if Spy has a later time stamp than Tether?

The relevant rules for timestamps are 613.6 and its subrules. As laid out there, the effect from a static ability usually has the same timestamp as the object it's on, unless the ability was given to that object by some other effect, in which case the timestamp is the same as that of the effect that created the ability. Skatewing Spy's ability was not granted to the Spy by some other effect, so the timestamp of its effect is the same as the Spy's itself.

While there are actions that will cause a permanent to receive a new timestamp (such as turning it face down or face up–see 613.6 for a full list), putting a counter on some other object is not one of them. Putting a counter on Skitter Eel doesn't change the timestamp of Skatewing Spy, so it doesn't change the timestamp of the Spy's static ability either, and the Eel will only have flying if Skatewing Spy already has a later timestamp than Sky Tether.