Double oven lost all power including accessories
The fourth wire (for both subpanel and oven) has everything to do with safety.  It will not impede an appliance from functioning.  

What will break the appliance is certain wires breaking or losing contact.  Generally on ovens, the heaters are 240V (they use hot and hot) and the controls are 120V (they use one hot and neutral).  So when the controls go out, that calls into question one hot and neutral.

Losing one of the hots will only knock out the heating elements.  

Losing the other hot will will cause everything to stop working if it is the particular hot that the 120V controls are using.  This is one of your possibilities. 

Losing the neutral will cause everything to stop working.  (the heating elements could turn on if the controls worked, but the controls don't because they need neutral).   However, on a 3-wire oven, losing the neutral has a second problem: the controls are now seeking to return current via neutral, meaning they are "pulling neutral up to 120V".  That is a normal and harmless failure mode, except you bonded neutral and ground in the oven.  Which means this effect is also pulling the chassis of the oven to 120V, energizing it.  If you touch the oven and anything which is grounded, you will get shocked.  This is why we don't like 3-wire oven connections. 

On the 3-wire subpanel, the same problem exists throughout the subpanel-served circuits.  If the neutral between subpanel and main panel has a problem, then similarly, it will light up every "ground" fed off the subpanel.  GFCI's can help reduce this risk.  But the most complete is retrofit a ground wire between main and subpanel, and then separate neutrals and grounds in the sub. (panel manufacturers sell separate ground bars, and you can detach neutral from the chassis of the panel).  Recent Electrical Codes have given very liberal permission to retrofit grounds.  You could also retrofit a ground to the oven and convert it to 4-wire. 

So I would first start by checking loads on both poles of the subpanel. If one pole is good and the other bad, it is a bad hot wire before the subpanel.  If both sides have weird voltages (which add up to 240V) it's a bad neutral before the subpanel (dangerous!) If all other loads are good, then check hot and neutral connections both at the sub and at the oven. Be wary of the oven's chassis being potentially energized.