a typical single oven, drawing less than 3kW (like yours) is safe on an ordinary plug and socket, with a 13A fuse.
The damage round the neutral pin of your plug suggests that when originally fitted, the wire was not firmly gripped in the pin-top terminal (the brass screw is rather small and difficult to get really tight). This looseness will have given a poor connection, which will have overheated, and the heat will have made the connection worse, as the brass terminal and copper wire will erode and eventually burn away. This will make it get even hotter, and the cycle continues until the wire burns away, or the house burns down, or you fit a new plug.
Copper wire which has been overheated becomes softer, and squashes away over time when you try to make a tight connection, so any copper that looks darker must not be re-used. The insulation will also become brittle and prematurely aged from heat.
Cut away or replace all the heat-marked flex; use a new plug by a good brand like MK, Crabtree or MEM; and replace the socket as well. If the wiring inside the socket shows evidence of damage, it will have to be replaced or repaired. Unless you are equipped to do this, it will probably be more economical to engage a qualified, well-recommended local electrician than to buy the correct tool and learn to use it.
It is preferable to have a large, fixed appliance like an oven on its own dedicated circuit, rather than on the general house ring, and when a kitchen is built or updated, this is worth considering, especially if you expect to have numerous large appliances, or if you think that in future you might want a double oven, or an electric cooker. Kitchen fitters are not usually good electricians.
too slow!
but very thorough!