Why is changing data types not effecting the database size?
How are you measuring the size of your tables and indexes?
The way SQL Server works is once the database is allocated disk space, it still retains that disk space and only marks it empty / for re-use internally when data is reduced / removed until a SHRINK or similar operation occurs that forces the release of that acquired space back to the disk.
So changing data types of a fixed-width to a type that uses less space doesn't automatically trigger the release of that previously claimed space back to the disk unless you rebuild your indexes, run a SHRINK, etc.
As Martin points out in the comments, if you are using compression, then likely the used space of the BIGINT column that only held INT values was already being compressed to an equivalent space use of an INT column, which would also explain why you wouldn't see it change in size by changing data types, even if you did rebuild your indexes or ran a SHRINK operation.