The red phenotype was noted in Scots by the Romans--so this is of course long before any viking invasions. So too, the dark 'spanish looking 'Welshman and the fair,lanky Englishman-noted centuries before any saxons/vikings came to England.
There is no such thing as 'Roman' dna. The romans in Britain were not all Italians-they were Roman citizens who came from the far reaches of the Empire. They could have been German,Spanish, African etc. they were also not settlers, they were soldiers. they hadn't come to have kids & farm the land, they colonized & ruled, to bring the wealth of Britannia to the Empire. Geneticists say any genetic legacy they left was just a drop in the dna sea.
I actually know more brown haired people in England (where I live ) than natural blonds, though blue eyes are common,and the hair colour is more 'middling'-nowhere near as dark as that found in the 'celtic' countries. that said, there is LOTS of pre-saxon dna in the English,particularly in the female lines.in fact in some very 'English' counties such AS Oxfordfordshire, the population is more 'celtic' than anything else.
I actually know MANY scots who are not red or fair haired, but dark/black haired,often with blue eyes-like many of the Irish-not surprising,since the Scoti were an irish tribe.there were of course people living in Scotland before the Scoti arrived-again, the prevalent group overall is of the basque dna type which must have arrived after the last ice Age.
Infact this group, is prevalent all over Britain INCLUDING England - (my brown haired,hazel eyed, English-on-all-sides partner has mtdna H1.)
I myself am of Irish descent and my grandmother was black haired, olive skinned with very large dark eyes. She looked Spanish or Italian. i'm similar but somewhat lighter. our mtdna is T2,which came out of the near East,then split, one group going to North East, another line West along the Med. I would guess my ancestor probably came from the latter.
Anyway, putting it simply, probably most of the early (mesolithic/neolithic) settlers of Britain & Ireland had varying shades of dark hair. By their bone structure,they were what we called mediteranean types .Some may have had red hair even then (you can find it in Spain too) Most probably had brown eyes but the blue gene surfaced somewhere around 10,000 BC so there were probably variations that were bred into the population over time.