How to make LAG() ignore NULLS in SQL Server?

Does anyone know how to replace nulls in a column with a string until it hits a new string then that string replaces all null values below it? I have a column that looks like this

Original Column:

PAST_DUE_COL           
91 or more days pastdue        
Null
Null
61-90 days past due          
Null
Null
31-60 days past due
Null
0-30 days past due
Null       
Null
Null            

Expected Result Column:

PAST_DUE_COL           
91 or more days past due        
91 or more days past due
91 or more days past due
61-90 days past due          
61-90 days past due 
61-90 days past due 
31-60 days past due
31-60 days past due
0-30 days past due
0-30 days past due      
0-30 days past due
0-30 days past due

Essentially I want the first string in the column to replace all null values below it until the next string. Then that string will replace all nulls below it until the next string and so on.

SQL Server 2022 added support for the ignore nulls option in window functions: yay!

We can just use last_value():

select t.*,
    last_value(past_due_col) 
        ignore nulls 
        over(order by id) new_past_due_col
from mytable t

Demo on DB Fiddle.

In earlier versions (< 2022), where SQL Server window functions do not support the ignore nulls option, we can work around with some gaps and island technique:

select t.*, max(past_due_col) over(partition by grp) new_past_due_col
from (
    select t.*, count(past_due_col) over(order by id) grp
    from mytable t
) t

The subquery does a window count that increments every time a non null value is found: this defines groups of rows that contain a non-null value followed by null values.

Then, the outer uses a window max() to retrieve the (only) non-null value in each group.

This assumes that a column can be used to order the records (I called it id).

Demo on DB Fiddle:

ID | PAST_DUE_COL            | grp | new_past_due_col       
-: | :---------------------- | --: | :----------------------
 1 | 91 or more days pastdue |   1 | 91 or more days pastdue
 2 | null                    |   1 | 91 or more days pastdue
 3 | null                    |   1 | 91 or more days pastdue
 4 | 61-90 days past due     |   2 | 61-90 days past due    
 5 | null                    |   2 | 61-90 days past due    
 6 | null                    |   2 | 61-90 days past due    
 7 | 31-60 days past due     |   3 | 31-60 days past due    
 8 | null                    |   3 | 31-60 days past due    
 9 | 0-30 days past due      |   4 | 0-30 days past due     
10 | null                    |   4 | 0-30 days past due     
11 | null                    |   4 | 0-30 days past due     
12 | null                    |   4 | 0-30 days past due