You can use the following:
SELECT *, REPLACE(column_name, '_', '') AS new_value
FROM table_name 
WHERE REPLACE(column_name, '_', '') LIKE '120515%';
demo on dbfiddle.uk
You can't use the column alias on the WHERE part of your query because the SELECT part is executed after the WHERE part. See the following StackOverflow answer describing the query execution order of MySQL.
So you have to use the REPLACE on the WHERE part. In case you don't need to see / use the new_value you can remove the REPLACE on SELECT.
In case the format is always xxxx_xx_xx, you can also use a solution like the following:
SELECT *
FROM table_name 
WHERE column_name LIKE CONCAT(INSERT(INSERT('120515', 5, 0, '_'), 8, 0, '_'), '%');
demo on dbfiddle.uk
As @Madhur Bhaiya already mentioned in the comments, you can use a index on the column_name to improve the performance on a large table.