How to assign day of the week to different tills
As mentined in my comment, you need an order to things especially when  ypou want to use row_number, the it is demanded.
The query does in principle what you want, but as there is no order to your table, or outcome, the result like you want id not possible.


CREATE TABLE [TillNumbersAssignedtoStaff](
    [AssignedWeek] date NULL,
    [StaffID] int NULL,
    [TillNumbers] varchar(3) NOT NULL
    
) 



GO


✓




INSERT INTO [dbo].[TillNumbersAssignedtoStaff]
VALUES ('2020-10-19',190,'AB1')
,('2020-10-19',190,'AB1')
,('2020-10-19',190,'AB3')
,('2020-10-19',190,'AB5')
,('2020-10-19',190,'AB3')
,('2020-10-19',190,'AB4')
,('2020-10-19',190,'AB2')
,('2020-10-19',190,'AB3')
,('2020-10-19',190,'AB2')
,('2020-10-19',180,'BY5')
,('2020-10-19',180,'BA3')
,('2020-10-19',180,'BY4')
,('2020-10-19',180,'BA1')
,('2020-10-19',180,'BY2')
,('2020-10-19',180,'BE2')
GO


15 rows affected




WITH X AS (
SELECT *
     ,ROW_NUMBER() OVER (PARTITION BY [StaffID] ORDER BY [AssignedWeek],[StaffID]) rn 
FROM TillNumbersAssignedtoStaff
 )
 SELECT [AssignedWeek],[StaffID],[TillNumbers]
      ,ROW_NUMBER() OVER (PARTITION BY (X.rn % 2),[StaffID] ORDER BY rn ASC) rn 
FROM X
ORDER BY [StaffID]
GO


AssignedWeek | StaffID | TillNumbers | rn
:----------- | ------: | :---------- | -:
2020-10-19   |     180 | BA3         |  1
2020-10-19   |     180 | BA1         |  2
2020-10-19   |     180 | BE2         |  3
2020-10-19   |     180 | BY5         |  1
2020-10-19   |     180 | BY4         |  2
2020-10-19   |     180 | BY2         |  3
2020-10-19   |     190 | AB1         |  1
2020-10-19   |     190 | AB5         |  2
2020-10-19   |     190 | AB4         |  3
2020-10-19   |     190 | AB3         |  4
2020-10-19   |     190 | AB1         |  1
2020-10-19   |     190 | AB3         |  2
2020-10-19   |     190 | AB3         |  3
2020-10-19   |     190 | AB2         |  4
2020-10-19   |     190 | AB2         |  5


db&lt;&gt;fiddle here