ALTER TABLE on dependent column

I believe that you will have to drop the foreign key constraints first. Then update all of the appropriate tables and remap them as they were.

ALTER TABLE [dbo.Details_tbl] DROP CONSTRAINT [FK_Details_tbl_User_tbl];
-- Perform more appropriate alters
ALTER TABLE [dbo.Details_tbl] ADD FOREIGN KEY (FK_Details_tbl_User_tbl) 
    REFERENCES User_tbl(appId);
-- Perform all appropriate alters to bring the key constraints back

However, unless memory is a really big issue, I would keep the identity as an INT. Unless you are 100% positive that your keys will never grow past the TINYINT restraints. Just a word of caution :)

If your constraint is on a user type, then don't forget to see if there is a Default Constraint, usually something like DF__TableName__ColumnName__6BAEFA67, if so then you will need to drop the Default Constraint, like this:

ALTER TABLE TableName DROP CONSTRAINT [DF__TableName__ColumnName__6BAEFA67]

For more info see the comments by the brilliant Aaron Bertrand on this answer.