Answer
Yes, the use of nonrelativistic formulas are justified in the Bohr atom.
The value of $\sqrt(1-v^2/c^2)$ is 0.99997
Work Step by Step
The velocity of electron in ground state of hydrogen atom is given by-
$v=h/2\pi mr_1
=6.63*10^{-34}/[2\pi *(9.1* 10^{-31})*(0.529*10^{-10})]
=0.022*10^8 m/s$
so, $v/c=(0.022*10^8)/(3*10^8)=7.33*10^{-3}<<1$
so, the use of nonrelativistic formulas are justified in the Bohr atom.
The value of $\sqrt(1-v^2/c^2)$= $\sqrt 1-(7.33*10^{-3})^2$ = 0.99997
