Couple of points
It's not the "fuse box"
It's the supply service head.
Old ones were often cast-iron, sometimes bakelite, and after being connected up, were often filled with molten pitch, which is an insulator and sets hard, perhaps intended to prevent deterioration or tampering.
When unusually hot, the pitch melts.
The local electricity network operator will attend and change it for a new one. You don't need a private electrician. There will not be a charge unless you have damaged it or been running a bootleg cannabis plantation.
It is usually caused by a prolonged overload. at this time of year it might be electric showers or air conditioning.
It's not especially worrying. The phone no of your DNO will be at the bottom of your electricity bill. Your supplier will probably not own or maintain the network. It needs a trained operative with protective clothing and protection for his face and hands, as the power is permanently on, and effectively unfused.
If you can't find an electricity bill, you can phone the supplier, who should know and/or pass the message on.
Your incoming electricity cable look to me as if it's about 90 years old. However it is probably perfectly safe. The engineer who changes the service head may strop back the armour to see.