Dates can be stored in various formats in the SF org according to user's preference & country. Though you see the date in your preferred format, it will get fetched with the code in standard format of "YYYY-MM-DD" only (it gets automatically converted to the above mentioned format and It can not be changed).
but there is a work around with which you can access date field in required format in SMS Template. The workaround is-
You can create new formula field on the same object of type text
Formula value should be - TEXT( DAY(Date Field)) & "/" & TEXT(MONTH(Date Field)) & "/"& TEXT(YEAR(Date Field))
This field will store the text version of Date Field in dd/mm/yyyy format.
- Then you can use this formula field in SMS Template in place of Date field and it will work as per your requirement.