The saving grace to many such installations is that in the same way the MCB takes time to heat up and trip, so does the wiring take time to heat up and do damage, and the time constants in the breaker are such that the worst you will normally do is have to change the breaker one day when it trips and refuses to reset, the wiring, which is what causes the fire if it overheats is normally OK, unless actually under-size or terminated badly.
Over running cable is not good, but the effect of small overloads (I mean temperatures below PVC melting point, perhaps 90-105c instead of the design temperature of 70, corresponding to perhaps 150% of the normal temperature rise, and power dissipation ) is not an instant and catastrophic failure , but more a premature aging of the cable. For this reason many under-specified installations will run on happily for many years being overloaded for perhaps an hour a day or so, whether advisable or not.
If you doubt this, consider that mercifully most broken ring mains are discovered by testing when later work is done, and not by the fire brigade investigation teams (and I fear there are plenty more broken rings out there waiting to be found).
However, there is no real excuse to install something from new as borderline, so please either use the next size up cable, (or next size down in breaker).