I have moved to a property with 5amp round sockets around the room. I want to remove the standard 3 pin plugs from my table lamps and re-wire them with the 5amp round pin plugs I have bought. This means they will be on a mains dimmer circuit, so I can choose to have the spotlights on, or the lamps come on with one switch. The problem is, 2 of the table lamps have 3amp fuses in. Can I safely attach them to a 5amp round pin plug and put them in the sockets without blowing anything up?
Sure. As long as the plug you're attaching is rated higher than the lamp itself, there's no issue. The lamps are fused at 3A, so they'll open far before pulling the rated capacity of the cord cap. Still, it wouldn't matter if they weren't fused.
What you can't do is attach a 3A end to a 5A lamp. The reverse, however, is fine. You can attach a 30A appliance receptacle to a radio if you wanted. I don't know why you would want to, but you're not creating an unsafe configuration by doing so.
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
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.
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
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.