This can be done as follows. Starting from your

ClearAll["Global`*"]
u[x, y] = y/Sqrt[-x + Sqrt[x^2 + y^2]];v[x, y] = (-x + (x^2 + y^2)^(1/2))^(1/2);

, we have

FullSimplify[ ComplexExpand[y/Sqrt[-x + Sqrt[x^2 + y^2]] + 
 I*(-x + (x^2 + y^2)^(1/2))^(1/2) /. {x -> r*Cos[\[Theta]], 
 y -> r*Sin[\[Theta]]}, Assumptions -> r > 0 && \[Theta] >= 0 && \[Theta] <= Pi]

Sqrt[2] E^((I \[Theta])/2) Sqrt[r]

and

Sqrt[2] E^((I \[Theta])/2) Sqrt[r] /. {r -> Abs[z], \[Theta] -> Arg[z]}

Sqrt[2] E^(1/2 I Arg[z]) Sqrt[Abs[z]]

in the upper complex half-plane. Making use of the Riemann-Schwarz principle with the positive ray of the real axis as $\gamma$, we draw the conclusion this is valid in the whole complex plane with the cut along the negative ray of the real axis.

This can be done as follows. Starting from your

ClearAll["Global`*"]
u[x, y] = y/Sqrt[-x + Sqrt[x^2 + y^2]];v[x, y] = (-x + (x^2 + y^2)^(1/2))^(1/2);

, we have

FullSimplify[ ComplexExpand[y/Sqrt[-x + Sqrt[x^2 + y^2]] + 
 I*(-x + (x^2 + y^2)^(1/2))^(1/2) /. {x -> r*Cos[\[Theta]], 
 y -> r*Sin[\[Theta]]}], Assumptions -> r > 0 && \[Theta] >= 0 && \[Theta] <= Pi]

Sqrt[2] E^((I \[Theta])/2) Sqrt[r]

and

Sqrt[2] E^((I \[Theta])/2) Sqrt[r] /. {r -> Abs[z], \[Theta] -> Arg[z]}

Sqrt[2] E^(1/2 I Arg[z]) Sqrt[Abs[z]]

in the upper complex half-plane. Making use of the Riemann-Schwarz principle with the positive ray of the real axis as $\gamma$, we draw the conclusion this is valid in the whole complex plane with the cut along the negative ray of the real axis.