Field due to coil 1: $B_1 = \frac{\mu_0 I}{2R}$ (along, say, x-axis)
Field due to coil 2: $B_2 = \frac{\mu_0 \cdot 2I}{2R} = \frac{\mu_0 I}{R}$ (along y-axis, since planes are perpendicular)

The two fields are perpendicular to each other:
$$B_{net} = \sqrt{B_1^2 + B_2^2} = \sqrt{\left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{R}\right)^2}$$
$$B_{net} = \frac{\mu_0 I}{R}\sqrt{\frac{1}{4} + 1} = \frac{\mu_0 I}{R} \cdot \frac{\sqrt{5}}{2} = \frac{\sqrt{5}\,\mu_0 I}{2R}$$