Can I overstay a UK visa? Can I apply for an extension?

Or is it not allowed to stay after the days of visa expiry?

It is not allowed. Every UK entry clearance has an expiration date. You need to vacate the UK by midnight of that date. If you remain in the UK beyond the expiry, you become an overstayer.

To overstay a UK visa is seen by the authorities as full on abuse and they will get very upset with you if you get caught. Indeed very upset. They are reflecting the current British public mood which has become soured and angry over media stories highlighting abuse. So if you get caught inland, it's off to Yarl's Wood courtesy of HM. I used to go there and can attest it is abysmal.

So don't do it. Tell your father to change his flight.

Now to your other question, a bit more interesting...

What should be done in order to exceed about 5 days of expiry date ?

The law has a little known provision for doing this. It's stated in Paragraph 31A of the Immigration Rules, it's called "Variation of leave to enter or remain in the United Kingdom"...

31A. Where a person has arrived in the United Kingdom with leave to enter or remain in the United Kingdom which is in force but was given to him before his arrival, he may apply, on arrival at the port of entry in the United Kingdom, for variation of that leave. An Immigration Officer acting on behalf of the Secretary of State may vary the leave at the port of entry but is not obliged to consider an application for variation made at the port of entry.

What this means is that Immigration Officers (IO's) are empowered to do anything they want with respect to entry clearances. And so a person can explain their situation and request a variation during their landing interview. The outcome of this is solely down to personal impact and articulation skills. If successful (0% chance of this), he will endorse a variation on the entry clearance. If the IO gets upset (and that's 100% guaranteed unless you are Dale Carnegie or Kim Kardashian), he can cancel your father's entry clearance and issue him a removal notice.

Nothing prevents a person from trying to benefit from Paragraph 31A (I have seen it work), but things can quickly devolve into an all-or-nothing gamble.

So to answer your question, yes there is a provision in the rules. And 'best practices' advice is don't try it.

My uncle has a visa valid for six month which was issued in April. And he is intending to visit UK in around 1st of October which is apparently 10 days beforehand of expiry date.

Read it VERY carefully.

The visa is valid for six months from its start date or the stay is valid for six months from date of arrival ? Visitor visas can be issued for start dates up to 90 days in advance of the issue date.

If the first one, your uncle has six months from April to enter the UK. Then he is allowed to stay for a certain period of time.

I suspect this is the case, because Nepalese nationals require an entry clearance before they depart.