I found Charles' answer (and the linked workaround in the comment by maxcastaneda) very useful. I followed this approach and it works for me. I have streamlined it a bit and created the following query that brings up all required synonyms to create.
As a prerequisite for this snippet both the original DB and the synonym/alias db have to be on the same server otherwise in case you use linked server or so you have to modify it a bit.
It should be fairly easy to put this into a small sp to update the synonyms automatically.
USE <SYNONYMDB>
SELECT
'[' + TABLE_NAME + ']',
'[' + TABLE_SCHEMA + '].[' + TABLE_NAME + ']',
'IF EXISTS (SELECT * FROM sys.synonyms WHERE name = ''' + TABLE_NAME + ''') DROP SYNONYM ['+ TABLE_NAME + ']; CREATE SYNONYM [' + TABLE_NAME + '] FOR <ORIGINALDB>.' + TABLE_SCHEMA + '.[' + TABLE_NAME + ']' AS SynonymUpdateScript FROM <ORIGINALDB>.INFORMATION_SCHEMA.TABLES
Don't forget to enter you Db names at the <...> spots.
Just copy the content of the SynonymUpdateScript
Column and execute it in the synonym DB - or create a stored procedure for this task.
Be aware there is an issue if you have views in place that refer to tables or other db objects without the 2 part naming convention. Those synonyms won't work. You should fix this in the original objects / views.