Can I rewire an audio amplifier with a round pin plug to fit sockets in living room made for lamps?
Can I wire a round pin 5 amp plug onto the amp


Nothing stops you from doing so. 


  Or does an adapter exist


Yes, adapters exist, they are sold as travel adapters for people from the UK who are visiting India (the indian 6A plug is equivilent to the 5A BS546).

Unfortunately many travel adapters are frankly complete garbage. I belive https://www.amazon.co.uk/Go-Travel-Earthed-Compact-Convertor/dp/B079VRTCRX is a non-garbage one but I haven't bought one personally.



Be aware that 5A sockets found in a domestic premesis will almost certainly be fed from a lighting circuit. So pay attention to total load on said circuit. They will also likely be fed via a light switch, people may find it confusing if the hi-fi turns off when they turn off the light switches.

Also make sure the socket is not fed via a dimmer. If it is then you would need to change the dimmer for a conventional switch before plugging anything other than dimmer-compatible lighting into it.