Partition Function COUNT() OVER possible using DISTINCT

There is a very simple solution using dense_rank()

dense_rank() over (partition by [Mth] order by [UserAccountKey]) 
+ dense_rank() over (partition by [Mth] order by [UserAccountKey] desc) 
- 1

This will give you exactly what you were asking for: The number of distinct UserAccountKeys within each month.

Necromancing:

It's relativiely simple to emulate a COUNT DISTINCT over PARTITION BY with MAX via DENSE_RANK:

;WITH baseTable AS
(
    SELECT 'RM1' AS RM, 'ADR1' AS ADR
    UNION ALL SELECT 'RM1' AS RM, 'ADR1' AS ADR
    UNION ALL SELECT 'RM2' AS RM, 'ADR1' AS ADR
    UNION ALL SELECT 'RM2' AS RM, 'ADR2' AS ADR
    UNION ALL SELECT 'RM2' AS RM, 'ADR2' AS ADR
    UNION ALL SELECT 'RM2' AS RM, 'ADR3' AS ADR
    UNION ALL SELECT 'RM3' AS RM, 'ADR1' AS ADR
    UNION ALL SELECT 'RM2' AS RM, 'ADR1' AS ADR
    UNION ALL SELECT 'RM3' AS RM, 'ADR1' AS ADR
    UNION ALL SELECT 'RM3' AS RM, 'ADR2' AS ADR
)
,CTE AS
(
    SELECT RM, ADR, DENSE_RANK() OVER(PARTITION BY RM ORDER BY ADR) AS dr 
    FROM baseTable
)
SELECT
     RM
    ,ADR

    ,COUNT(CTE.ADR) OVER (PARTITION BY CTE.RM ORDER BY ADR) AS cnt1 
    ,COUNT(CTE.ADR) OVER (PARTITION BY CTE.RM) AS cnt2 
    -- Not supported
    --,COUNT(DISTINCT CTE.ADR) OVER (PARTITION BY CTE.RM ORDER BY CTE.ADR) AS cntDist
    ,MAX(CTE.dr) OVER (PARTITION BY CTE.RM ORDER BY CTE.RM) AS cntDistEmu 
FROM CTE

Note:
This assumes the fields in question are NON-nullable fields.
If there is one or more NULL-entries in the fields, you need to subtract 1.