This is a bit of a guess but it's based upon similar experiences.
Nearly every Ford I've had has (sooner or later) had a weird problem with the radio whereby it won't work for the first 10 minutes or so of a journey but then comes on normally thereafter. (That's particularly true when it's cold and/or damp and when the lights or wipers are on). It seems that although the battery has plenty of current to start the engine, the actual voltage across its terminals is initially too low to operate the radio. Once the alternator has had sufficient time to boost the battery everything returns to normal.
My present Ford has exhibited that problem for several years (despite the battery always starting the engine first time) but replacing the battery on previous cars has always fixed the problem.
So, even though your car's not a Ford and the problem relates to different electrical components, I still suspect a battery problem.