Multiple records and determine if a combination if values exist and write this to a new column
For an Update to the existing table, one solution would be:

UPDATE t
SET t.New_Column = CASE WHEN types.mintype = 'F' and types.maxtype = 'F' THEN 'FTW'
                        WHEN types.mintype = 'N' and types.maxtype = 'N' THEN 'Non-FTW'
                        WHEN types.mintype = 'F' AND types.maxtype = 'N' THEN 'Mixed' END
FROM tbl t
INNER JOIN 
(SELECT ORDR, 
       MIN(TYPE) mintype,
       MAX(TYPE) maxtype
FROM tbl
GROUP BY ORDR) types
    ON t.ORDR = types.ORDR


Though, I would also, as Chris Aldrich said in the comments, suggest using a view. That view could be:

CREATE VIEW v_tbl
AS 
SELECT t.ORDR, 
       t.TYPE,
       CASE WHEN types.mintype = 'F' and types.maxtype = 'F' THEN 'FTW'
            WHEN types.mintype = 'N' and types.maxtype = 'N' THEN 'Non-FTW'
            WHEN types.mintype = 'F' AND types.maxtype = 'N' THEN 'Mixed' END New_Column
FROM tbl t
INNER JOIN 
(SELECT ORDR, 
       MIN(TYPE) mintype,
       MAX(TYPE) maxtype
FROM tbl
GROUP BY ORDR) types
    ON t.ORDR = types.ORDR