Thinking of going overseas to study overseas, either US or UK.

Does anyone who has knowledge of civil service/stat board recruitment processes know if the SG government care about reputation/ranking of universities?

I know that the usual suspects like Oxbridge, Yale, Harvard, LSE etc. are looked upon favorably but what about below those? Is there a difference in the mind of the public service recruiters between a quite good but not top university and a mid-tier university? For example, universities like Manchester, Edinburgh, Bristol etc...they are quite prestigious in UK but wondering if there is any 'reputation effect' in the eyes of the govt employers here.