My current Schengen visa expires on 06/04/2018, and I intend to travel to Belgium on 17/04/2018. I submitted my application today. Does the fact that I still have a valid visa mean that the embassy cannot begin to look at my application until after the 6th? I’m worried the visa may not arrive in time as it can take up to 10 working days.
Since the visa you already have will not be valid for the trip you're planning, it does not prevent the issuance of a visa that will be valid on your proposed travel dates.
There's no rule saying that holders of a currently valid visa cannot apply for a necessary visa for a visit in the future.