So did I. I thought the vibrations were passing through the circulation pump into the pipes, but this resonance thing seems to happen (to a lesser extent) even when the pump is off.

Does it disappear when the heat pump is off?  What about when the fan and water pump on the heat pump are running, but not the compressor(which will happen at times during cycling).

Are you absolutely sure it's the heat pump.  Sorry to ask!  Once one becomes tuned into noises they can become more obvious than they were previously.  I didn't hear my dishwasher at night until I got a heat pump, then, presumably because I was listening for noise in the early days, I started to hear it!  Weird psychology I know, but nevertheless the case.  I live in a fairly quiet area but still at night there is, depending on wind direction, a low pitched hum from somewhere, possibly just distant road noise filtered by the fabric.  I'm more conscious of that than I was pretty heat pump!

I would focus on the original source of the noise which I thought you said was the pump.  Is it?

So did I. I thought the vibrations were passing through the circulation pump into the pipes, but this resonance thing seems to happen (to a lesser extent) even when the pump is off.

It's also very subtle, the engineers who came were saying they couldn't hear it. Even I can barely hear it, but the problem is that when it's all quiet at night it makes it very hard to sleep, and when I'm in my study for a couple of hours, my ear starts to hurt really badly, and then hurts for hours after leaving the study.

Octopus replaced my old pump :

With a smaller one like this:

They even put in some kind of rubber bracket to reduce the vibration.
I thought it made a difference, but maybe it was a placebo effect, cos it's now noisy again.

I am at a loss, I don't know where to look for the source of the noise. 

I'll read the article and try to undo the balancing right now.

Balancing can be a long process (it can also be very quick especially if the settings are almost right) and its possible to mess things up if you turn down LSVs too much.  If you are happy with the temperature I advise you to leave it unless you have positive reason to suspect its causing a problem.  If you wanted to you could, without changing anything, check that at least one radiator has its LSV fully open.  It would usually be the one furthest from the heat pump or one nearby.   Its important that at least one LSV is fully open and everything else is balanced to that, otherwise you are creating unnecessary restrictions which compromise performance and increase noise. 

Elbows in pipework may exacerbate noise, but are unlikely to be the cause unless the radiators are so poorly balanced that a disproportionate flow is going through one or two rads.    But if this were the case you would notice it in the heating characteristics.  If it is causing it here, the noise will be especially apparent.  Is it.

I would focus on the original source of the noise which I thought you said was the pump.  Is it?

I'll read the article and try to undo the balancing right now.

Also, do you think a big, super wide radiator with a bend like this could be causing some of the noise?

This is the hallway radiator next to the living room.

2. Since the balancing, a room in the house which was previously very cold at night, is now quite warm.
3.The radiator in this room has not been changed, it's still the small raditor from before the HP install.
4. How is this achieved?

By opening up the lockshield valve on the radiator in question, or closing down a bit lockshield valves on other radiators (the lockshield valve, often abbreviated to LSV, is the valve opposite the TRV).  This forces more water to flow through the radiator.  Thus for any given output the temperature drop (delatT) across the radiator reduces, its average temperature therefore increases and thus its output increases.

Thats what balancing is.  If you want to read up on it there is an article here.

I ask because the ringing/resonance I've been complaining about has gotten worse since the balancing was done.

Thats unfortunate.  It would be useful to know flow rate before and after as it might tell us something.  Changing the balancing is not the way to fix it however, unless somebody has closed down LSVs too much.

Have you tracked down definitely that its coming from the secondary pump.  Are all your TRVs fully open (ie set to max) and if not what happens to the noise if they are?

My strong advice to you is to stop clutching at straws and knuckle down to some detective work.  You are much more likely to track it down that any visiting engineer is, because you are there permanently.  Once its pinpointed and the characteristics clear in terms of what stimulates it or suppresses it, fixing it will be easier.  Im not saying that tracking it down will be easy, but you have a better chance than anyone else.

I have no idea. I suspect it was down to the HP engineers and the quality of the installation.

Since installing the HP our water hammer has gone, so I'd agree that replacing your HP with a gas boiler may not solve this.