Can I modify 3 ampere table lamps to work on 5 ampere round pin plugs?
3A lamp on 5A outlet

You should be OK to use a 3A appliance on a 5A circuit. The circuit is capable of supplying up to 5A but it is the individual appliance (lamp in this case) that determines how much current is drawn.

Round-pin plugs vs rectangular-pin plugs

Modern usage

I'm surprised a new home would have BS 546 round-pin plugs instead of the more usual BS 1363 rectangular-pin plugs.

Wikipedia says


  Although BS 546 plugs and sockets are still permitted in the U.K. and are used for special purposes, new installations since 1947 generally have used BS 1363 sockets and fused plugs.


Until now, I thought the only place where round-pin plugs were used was in theatrical stages, where I think they are used to prevent people plugging inappropriate devices into theatrical lighting circuits.

Safety

If the BS 546 plugs don't have built-in fuses, they don't provide any protection to the line-cord between the plug and the lamp.

BS 1363 plugs have additional safety features which may not be present in BS 546 plugs


fuse to prevent overload of appliance cord (&amp; hence fire.)
shutter in socket opened by earth pin (safer sockets).
insulating sleeves around base of live and neutral pins.
high insertion force to reduce risk of partial removal.
designed so earth wire is last to break under cord strain.


For example a small child can poke their hands (or part of a metal toy) behind a loose BS 546 plug and be in contact with 230 V.