Actually the singularity handler "DuffyCoordinates"
is enough.
NIntegrate[I62, {z, 0, Infinity}, {u, 0, Infinity}, {v, 0, Infinity},
Method -> {"GlobalAdaptive", "SingularityHandler" -> "DuffyCoordinates"},
PrecisionGoal -> 6] // AbsoluteTiming
(*
NIntegrate::slwcon: Numerical integration converging too slowly....
{3.011657`, 1.3744127589904662`}
*)
But to get greater precision, a higher working precision and more time are needed:
NIntegrate[I62, {z, 0, Infinity}, {u, 0, Infinity}, {v, 0, Infinity},
Method -> {"GlobalAdaptive", "SingularityHandler" -> "DuffyCoordinates"},
WorkingPrecision -> 24, PrecisionGoal -> 8] // AbsoluteTiming
(*
NIntegrate::slwcon: Numerical integration converging too slowly....
{70.5418, 1.37441274742384630555707}
*)
One can save a small amount of time by separating out the singularity at z == 0
with {z, 0, 1, Infinity}
in place of {z, 0, Infinity}
.