In England the date of the New Year was Lady Day, the 25th of March (not 21st) We still have a vestige of that in the tax and financial year start on 6th April, ie the same day adjusted for the loss of days at the change of calendar.
Pragmatically, when dealing with family history within a Country we don't normally start calculating what the date would be in the modern calendar, by subtracting days. It's normal to use the devise of double dating ie writing Jan 30th 1651/2 . If just one date is used, as is necessary in the data field on a profile, then conventionally one uses the latter date ie 1652 but then make it clear elsewhere ie in the bio what has been done. There can be a problem when dealing with events that take place in different countries because other European Countries changed their calendar 170 years earlier (see https://en.wikipedia.org/wiki/Old_Style_and_New_Style_dates for examples of conflicts)
What you have to be very careful with is transcribed or indexed dates .For example, in their catalogue, the National Archives has the PCC Wills catalogued with the probate dates in New style, Ancestry has the same wills with the probate dates as written (in this case you have the original image to check, though the probate date is not always easy to read and is written in Latin)
When you are working from an older English parish register it is usually quite clear that the year 'changes' in March. I believe that there are some parish registers that use double dates but I haven't seen one.
I am not always certain that there is any consistency on how datesfrom registers are written in the index in Ancestry or on Family search This is particularly so in user contributed pedigree/ancestral files on Family Search. I found this set of dates the other day, the first 9 are for the same child; some are in OS ie 1713, some NS ie 1714 Family Search
The other thing is that there is a difference between England and Scotland. James VI of Scotland changed the calendar in 1600 and although he became James I of England in 1603, the calendar was not changed until 1752. So 30th January 1651 written in a Scottish parish register means exactly that whereas over the border in Northumberland it would refer to a date a year later.
(just to add, in older English documents, including wills, you may find regnal dates ie on the 4th day of March in the 9th year of Henry VIII. There is a handy online calculator for those http://people.albion.edu/imacinnes/calendar/Regnal_Years.html )