I tried to prove it but i couldn't. But in my head i kept thinking that this integral gives the answer as pi/2 as the answer for all a. So i am kinda inclined to the assumption that it is indeed uniformly convergent but i cannot prove it using dirichlet's test or Abel's test or the Weirestrass M test for integrals.
PS :- Can anyone please give me the answer to this:- Is uniform convergence of an integral a necessary condition for differentiation under the integral sign?
It's not uniformly convergent, and it's not hard to see why: if it were, you could let $a\to0$ in $$I(a)=\int^\infty_0\frac{\sin a\,x}{x}\,dx$$ (because that's a sufficient condition for changing the order of limit and integration) and you'd have $I(a)\to I(0)=0$. But that's not true, as you know. Concerning your PS: no, it's not necessary, nor is it sufficient. If the integral you get by differentiation under the integral sign is uniformly convergent, too, that would be sufficient.