comments:
U/u V/v---I see different souces mean different things by case
Abs[r/(2 M) - 1] seems to solve the r>2 M & r < 2 M (inside/outside) cases
by U<->V swap
yes: (r/2M -1) Exp[r/2 M]=u^2-v^2 for boths cases due to a Sinh/Cosh switch
t: inside outside swap U<->V


V[r_, t_, M_] := Sqrt[Abs[r/(2 M) - 1]] * Exp[r/(4 M)] * Sinh[t/(4 M)]
U[r_, t_, M_] := Sqrt[Abs[r/(2 M) - 1]] * Exp[r/(4 M)] * Cosh[t/(4 M)]
r[U_, V_, M_] := 2 M (1 + ProductLog[(U^2 - V^2)/E])
t[U_, V_, M_] := 4 M ArcTanh[U/V]



ContourPlot[t[U,V,1],{U, -3, 3}, {V, -3, 3}, Contours -> {0,1,2,4}, ContourShading -> None,ContourStyle->Red]

ContourPlot[t[V,U,1],{U, -3, 3}, {V, -3, 3}, Contours -> {0,1,2,4}, ContourShading -> None,ContourStyle->Red]

ContourPlot[r[U,V,1],{U, -3, 3}, {V, -3, 3}, Contours -> {2.1, 2.5,3, 3.5}, ContourShading -> None]

ContourPlot[r[U,V,1],{U, -3, 3}, {V, -3, 3}, Contours -> {0.035,1.4,1.85}, ContourShading -> None]


Show[%,%%,%%%,%%%%]

 d={{D[U[r, t,1],r],D[U[r, t,1],t},{D[V[r, t,1],r],D[V[r, t,1],t}]
