See 'Years' here, to see that 1st January became the start of a new year, in the Roman calendar, in 153 BC
http://en.wikipedia.org/wiki/Roman_calendar
and here to confirm that the practice was continued, under the Julian calendar, from 45 BC:
http://en.wikipedia.org/wiki/Julian_calendar
Note particularly the section on 'New Year's Day' in that link. It shows that Samuel Pepys recognised 1st January as the start of a new year, even though he also accepted that the 'official' new year started on the 25th March.
Millions of people in this country still continue to recognise more than one date as the start of a new year. For example, there will probably be many Jews among those celebrating New Year on 31st December/1st January, despite the fact that they will have also celebrated Rosh Hashanah in September. Similarly, members of the Chinese community aren't barred from the celebrations on 31st December simply because they've got another new year starting on 23rd January. Such people (like Pepys) simply accept that the 'official' start of a year doesn't have to tie in with their own traditions.
Chris