As long as the file name is the only YYYYMMDDHHMMSS string in the file, I think you can just check for that pattern:

SELECT input, SUBSTRING(input, PATINDEX('%[2][01][0-9][0-9][01][0-9][012][0-9][0-5][0-9][0-5][0-9]%', input), 14) AS extract

FROM (

VALUES('\\cc0dware01\BMF\RMS\REBATERATES-WORKING-2016ToEnd08-20160905180112.csv'),

('\\cc0dware01\BMF\RMS\REBATERATES-WORKING-2016ToEnd08-20160905180113.xlsx'),

('c:\20160905180113.')

) AS test_data(input)

Edit: Added SQL code block.