I tested this on SQL Server 2012 SP1 on a local server (i.e. not Azure) and it also does not work. Therefore, most likely, you will have to abandon the index and recreate it on Azure.
ALTER TABLE Children ALTER COLUMN ChildName VARCHAR(50) NOT NULL
produced by:
Msg 5074, Level 16, State 1, Line 1 The index 'IX_Children_ChildName' is dependent on column 'ChildName'. Msg 4922, Level 16, State 9, Line 1 ALTER TABLE ALTER COLUMN ChildName failed because one or more objects access this column.
(Of course, you need to check that the column is not NULL, which would be an error even if there was no index in the column)
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