How to do ALTER TABLE TYPE efficiently
Instead of 


  Open each affected objects (i.e. ALTER SP) and execute it again so
  that the reference to table type will be updated


you can just find all the affected objects and refresh them using sp_refreshsqlmodule.

Here is the code to find the modules:

select object_name(referencing_id)
from sys.sql_expression_dependencies
where referenced_class_desc = 'TYPE' and referenced_entity_name =  'your_table_type_name';


To do refresh of all this objects you can simply generate the commands like this:

select 'sp_refreshsqlmodule ' + quotename( object_name(referencing_id), '''')
from sys.sql_expression_dependencies
where referenced_class_desc = 'TYPE' and referenced_entity_name =  'your_table_type_name';


Make the output to text instead of grid and copy + paste generated rows to SSMS and execute them.