The most efficient way would be to rectify the system and use high quality components to get the lowest losses possibly.
After you have rectified to DC, I would suggest using a very high efficiency buck converter.
So, rectify with high grade components to DC(focus is on low leakage here) then buck convert it to the power you need.
When you build your own system for lighting, that is using 120VAC and has high current output has the possibility for fire in failure modes. If someone is hurt or their property damage, you can probably be found liable, and possibly criminally negligent, depending on your country and their laws.
I recognize you said UK, but I will not pretend to know the laws there, I would suggest you spoke to an attorney about it if you are planning to switch your house to this. At minimum use Fuses everywhere they make sense and protect yourself from having a wiring fault and starting a fire.
The filter caps from the rectifier will have a large amount of power stored and a short after that point could easily become a fire if not protected against.
Improving on the Design
So, there is still more you can do to improve this, it just gets more complicated and will require control circuitry on your part. This control circuitry will require power, but it should be negligible with a good design compared to what you save. You will do this by having the LEDs blinking at a very high frequency with large off-time and over-current during the on-time.
LEDs have failure brought on by heating effects, this means there are many ways you can trick an LED. An LED Rated for 5A can be vastly overcurrent for a very short period of time without any damage done to the LED. The trick is to hit it with the current very quickly then have an off period where any heat is allowed to dissipate. The average light your eye sees will go up with lower power consumption.
I did a quick google search to attempt to give you a good example, but I failed. It seems the words involved in finding these methods are too often used in systems where things like overcurrent are a danger. You should play with it, I think you will find it surprising. I have actually done this myself, multiple times, and with excellent results. One of my professors made the duty cycle to long on purpose once to show me the changing of colors you can get in an LED due to heating also, don't do this, unless to laugh at it, because you have to bury your LED afterward.
I would suggest using a very simple 555 timer circuit. This circuit is driving a motor and uses a POT, but you could easily use 2 resistors of your choice and have it hard set to it's frequency. The motor is an example load and driving an LED is significantly easier(ie. inductance is much lower). This also adds a place to add an on/off switch. By placing a switch on the branch that is used for the "on" cycle and you can disable the system.
Efficiency When Off
Make sure when you leave a room and switch off all of the lights, or whatever unit it is that you feed from each Buck converter that you are also disconnecting the rectifier and the buck converter. They will continue to drain power when the load is disconnected. This will very quickly remove any power savings you have reaped.
You can read about this online, but LEDs really have the highest efficiency in low temperature applications, in room temperature you will probably not save any power over florescent lights at the same cost. You should be able to save a relative amount of power just by upgrading to high end components, but the system will cost you significantly more to implement.