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The simple answer is that it wasn't an issue because people at the time didn't use numbers to represent months; dates were written in full. We're so familiar with abbreviations and acronyms in text that we assume that they have always been in use, when they are actually comparatively recent.
When you say that "Great Britain and its colonies used March 25th as the beginning of the numbered year", that applied to the Civil or Legal Year. However, this was complicated by the use of regnal dating for official communications and legal documents, where the start of the year was based on the start of the monarch's reign. People were familiar enough with the dating systems to be able to convert between them.
As a modern parallel, the current British tax year starts on April 6th and is referred to by the years that it spans, so the tax year starting on 6th April 2016 would be 2016/17.