Tbar:=(1/(π^3))(∫_0^∞)\[DifferentialD]r(∫_0^1)(r (Sqrt[1-u^2])Cos[2 z r u-2δ(r u)])\[DifferentialD]u

For simple case of δ(r u)=0, the inner integral of Tbar becomes:

(∫_0^1)(Sqrt[1-x^2])Cos[-2 x]\[DifferentialD]x

This integral is in the form of (∫_0^1)f(x)Exp(i ω x) dx. For similar oscillatory integrals,
WLOG, let f(x)=Cos[x] and g(x)=x.  Cos[ω x] is the real part of Exp[i ω x]

a:=0

b:=1

n:=100

Here, n=ω x.

A:=1/Pi^3

f[x_]:=Cos[x]

g[x_]:=x

h[x_]=E^(I n* g[x])

J[x_]=(∫_a^b)h[x]*f[x]\[DifferentialD]x

Here, Act is the analytical integration.

Act=SetPrecision[A*Re[N[J[x]]],20]

T:=NIntegrate[h[x] f[x],{x,a,b}]

NInt=SetPrecision[A*Re[T],20]

Here, NInt is the numerical integration.

Iserles' method I(asymptotic):

Q1A:=((E^(n I) f[b]-f[a])/(n I))

Q2A:=((E^(n I) f[b]-f[a])/(n I))+((E^(n I) f'[b]-f'[a])/(n^2))

With higher order derivative terms:

Q3A:=((E^(n I) f[b]-f[a])/(n I))+((E^(n I) f'[b]-f'[a])/(n^2))-((E^(n I)f''[b]-f''[a])/(I n^3))

Q4A:=((E^(n I) f[b]-f[a])/(n I))+((E^(n I) f'[b]-f'[a])/(n^2))-((E^(n I)f''[b]-f''[a])/(I n^3))-((E^(n I) f'''[b]-f'''[a])/(n^4))

MI=SetPrecision[A*(Re[N[Q2A]]),20]

We only included the first derivative. One can include higher order derivatives as well.

Iserles' method II:

Q2F1:=(-(1/(n I))-6((1+E^(n I))/(n^3I))+12((1-E^(n I))/(n^4)))f[a]

Q2F2:=(((E^(n I))/(n I))+6((1+E^(n I))/(n^3 I))-12((1-E^(n I))/(n^4)))f[b]

Q2F3:=(-(1/(n^2))-2((2+E^(n I))/(n^3 I))+6((1-E^(n I))/(n^4)))f'[a]

Q2F4:=(((E^(n I))/(n^2))-2((1+E^(n I))/(n^3I))+6((1-E^( n I))/(n^4)))f'[b]

Q2F=Q2F1+Q2F2+Q2F3+Q2F4

MII=SetPrecision[A*(Re[N[Q2F]]),20]

Grid[{{Actual,NIntegrate,Asymptotic,IserlesII},{Act,NInt,MI,MII}},Frame->All]