Can someone please help with Electronics transient/capacitor question?
Your error is that you assume that the voltage after the switch, \$V_1\$ is \$\dfrac{V}{2}\$. That will indeed be the case in steady state, when the capacitor is fully charged. But that voltage varies! At the moment the switch closes the capacitor's \$R\$ and the right \$R\$ are parallel, so then  

\$ V_1 = \dfrac{R||R}{R||R + R}V = \dfrac{1}{3}V \$

Write down the three currents as a function of \$V_1\$, and since you know \$I_1\$ = \$I_2\$ + \$I_3\$ you should be able to find an equation for \$V_1\$.  

Note that since \$V_1\$ isn't constant you can't use the exponential curve for the capacitor's voltage. You'll have to write out a new integral.