Failed preamp would be my first guess.

Okay, so: I returned the pre-amp and got a replacement. 
 

Still the same issue - very loud hum and the music buzzes when it comes through. 
 

Thoroughly tested the cables and everything. 
 

Any more ideas?

DON’T play anything, but try plugging something like a CD player into the phono preamp to see if it hums. If it doesn’t you have confirmed the preamp and 300 input are good.

Take the turntable to somewhere you can plug it into a Phono input on a receiver and see if it hums there too. If it doesn’t it is good too.

Re-reading your original post on power cords, when on the same power strip did you try reversing just one plug in the socket at a time to see if the hum reduces. You’d need to work through all the combinations of plug directions.

Hi Stanley. Thanks for your help with this. 
 

I e tested various combinations of devices and am pretty happy that the 300 input connection is good, as is the turntable output. 
 

We are in the UK and have the three pin plug system so cannot reverse them - unless I am misunderstanding what this means. 
 

I wonder though - the amp has an earthbound in the plug but it’s a dummy and just plastic. As the Rega is grounded internally through the left put channel, could this be an issue?

“Grounds” are usually connected from a dedicated “Ground” terminal on the turntable to a dedicated “Ground” terminal on the preamp. Additional Ground connections from chassis to chassis can cause trouble.

Do you have hum if you disconnect the turntable while connecting the preamp output to Line-In? Likewise, if you skip the preamp and directly connect the turntable to Line-In, is there any hum (likely at a lower level)? Obviously you cannot listen to music with either of these connections. This is just an attempt to identify the hum source.

There is no hum when I connect the Rega straight into the line in on the 300. The sound isn’t great, but it is listenable.
 

no hum when just the amp plugged into the line in. The hum starts the second I power in the turntable.    
 

 Is the preamp power cable a three prong?  If so lift the ground with a cheater adapter if they have those in the UK.

As you perform my tests above, power the turntable. When you have the turntable connected normally, is there any change in the hum as you move the tonearm closer to the center of the record?

No, there is no discernible change in the hum as I live the tonearm. 

Okay, so problem solved. And you can all have a good chuckle at my expense. 
 

I was convinced I had bought a RP1. 
 

I in fact have a 1 Plus. Which has a built in Phono Stage. 

When in trouble shooting mode once you’ve decided that the problem must be e…] or cannot be …], you are likely to be blindsided.

In this case the problem was with the human. It’s good that you were open enough to investigate this.

Oh yea, double RIAA equalization can do that for you, close to 20 dB boost at 60 Hz.