DNA tests aren't that simple. Based on what I know about my family tree - and DNA tests taken by the Scottish side of my family - you'd have to cut me into pieces.
I'm supposedly half Scottish, half Slav. Turns out that there might be a smidgen of Polynesian and a smidgen of Latvian in the Scottish side. (History tells of some Tahitians who landed in St Andrews after a Scots merchant married a Tahitian woman and I've discovered that one of my great-grannies had a great-great grandpa who was a Latvian pastry cook who settled in Kirkcaldy.)
Dad's side is a mix of various Balkans nationalities, but apparently the family originated in Hungary and there might be a few other mixes in there, courtesy of the Roman Empire, Ottoman Empire and Austro-Hungarian Empire.
Having said all that...Dad came to the UK as a Displaced Person who'd fought with the Allies. `He was here twenty years before he could afford to pay naturalisation fees, etc.
However, he was obliged to carry an Alien Card until the early 1960s. He signed in at the local Police Station once a week and had to notify any change of address. He and the other men who came over at the same time were under no illusions about their situation: they knew that any bad behaviour meant that they'd be deported and they had no complaints about that.
I was at primary school when Dad underwent the naturalisation process. I recall that we were visited by local bobbies who also interviewed our neighbours to check whether we were troublesome. (Mum came from a long line of locals, with the exceptions of the Tahitian servant and pastry cook.)
The irony, actually, is that we had two feral (UK) families living in our block and when one of their sons smashed my front teeth (hence my single implant) my parents were scared to complain about it in case they made allegations about us and had Dad deported.
But yes - I agree that you when you live here, then you fit in. That was certainly Dad's view.
As for Travellers, I'm pretty certain that their DNA would just show them to be a Scottish/Irish/English mix.
I've mentioned on another thread that I had to stop a Traveller girl from assaulting a Romany girl one time. What sparked the Traveller girl's rage?
She was angry that the Romany children at the school where I worked referred to themselves as "Gypsies". According to the Traveller, only Travellers were gypsies.
We also had fights between Travellers and Carnival people and there were battles between Traveller adult men in a local Traveller site.
However, so far as I can see the toxic masculinity that causes young men to behave in a feral manner extends to nearly all groups in our area and it's become worse.
When I was a girl, only the lowest of the low would use his feet in a fight. By the time I retired, teenage boys were hunting in packs for lone males so that they could put the boot in. They thought that stomping on someone's head was funny.
The social media for the area seems to be full of pages of teenagers fighting at bus stations these days, to the extent that there are umpteen demands that free bus passes in Scotland be revoked.