<<<[How far does one have to go back before declaring (regionally or nationally) that something is non-native and should be eradicated?]>>>
According to Mr BING, Oaks, Beech, Holly and Spindle all arrived in around 4,000 BC, ie 6,000 years ago. See the BEECH in there

So how is beech not a native? I'm totally confused

I know there is this thing in the UK that Beech are not native trees, and clearly the woodland trust thinks it's not native. The last Ice Age finished about 10,000 years ago after around 100,000 years of ice cover, so ALL our flora and fauna have arrived here since then. It's so daft, isn't it? I love Beech trees, perhaps my favourites along with Scots Pine. I must investigate further.....
Tada!.....from BING again:<<<In recent years Beech has been considered to have somewhat of an identity crisis because scientists were unsure whether or not the species we find in Scotland was in-fact native. This is because Beech trees are naturally found only in southern parts of England. Those in Scotland – usually lining roads and avenues – were brought up in the Victorian period, and have colonised our countryside well ever since. Last year the experts finally discovered that the species growing in Scotland did originate from the UK and not mainland Europe >>>
So that's a bit late for the island beeches which have already been felled.