When we moved, we brought things two different ways. In containers, packed by a private company, and the fragile hand things we brought on the plane with us.
Besides the list that we gave for insurance reasons, the packers never asked what was recently purchased, and they took everything through customs, delivered it to us, etc, so we personally didn't speak or declare to Customs. I don't know what procedures they went through. There were a few very smallish items, like a couple of towels, as I recall, that had been newish, but since the tags were off and they had been used, they weren't questioned. Nor did I know whom to make the declaration to. Nothing on the forms indicated I had to declare anything, no one asked, etc. Customs do go through things, or at least they say they do...I didn't have an indication that anything was hand searched, so I don't know if everything is scanned, and only a few are hand searched, whatever. (They were tagged as having gone through customs, but of the few boxes I noticed, before the movers unpacked them, they were still sealed as when they left my house.)
The items we brought by hand we knew would be under scrutiny, so we went through declarations and explained it all to the customs officer. Certain items were antique, and fell within the non-taxable rules. Other things...well, this puzzled me, but maybe there was a rule about it I didn't know. We were quite open about everything. He let us pass, no problem, no paperwork, no extra tax. But we did explain it was part of the first time entering as a 'resident' and that other things were coming by container. There were questions about age of antiques, value of other things, total value that we carried. Trust me, it was well over the normal 'tourist' allowance, and I think it was important that we did say it was part of a move.
This was my own experience. And this was several years ago. Regulations could be increased due to the tense times.
I do know the latest thing when I crossed the Atlantic last month is that they are now asking everyone to unlock, and keep unlocked, their check-through luggage. I also noted our bags, as we collected them at Heathrow, came through with a green sticker. Signs on the wall identified those labels as being okay to take through 'nothing to declare'. Other stickers were shown in red, etc. telling people if their bags had those stickers to take luggage through declarations.
Frankly, we brought back lots of things that we'd bought on the trip. We'd fully expected to go through declarations. (Much of what we bought was for business and although we would have paid tax on them, we would also have claimed the VAT back...to us it would have just been paperwork.) My point is that we were prescreened and that may be a new policy. But, this is starting to get away from your question.
I can't answer for today's rules, but the above was my own experience. We were open about everything, and didn't have to pay any extra.