The minister has since been replaced.

1.这是一个简单句.
该句采用了现在完成时.
主语是The minister,系表结构be replaced.
since为副词,在句中作时间状语,意思是“自那以来/以后”.
2.可以把句子换写成：The minister has been replaced since.
= The minister has been replaced ever since.
= The minister has been replaced since then.
翻译：从那以后这个部长/大臣/牧师就被撤换了.