We are regularly asked for names of other customers using our products.
Do we require their permission to: (i) disclose their company name to others; (ii) to publish our customer names (this obviously excludes those we have NDA /Confidentiality agreements with)?
There is no rule of English law that generally prevents the publication or disclosure of corporate customers' names.
In particular cases, however, there may be restrictions on doing so. First, as you correctly identify, some NDAs and confidentiality clauses will prohibit the publication and/or disclosure of this information. Second, in some cases, the relationship between a services provider and customer will be inherently subject to confidentiality obligations (e.g. some medical and legal services providers). Third, the law of confidence can also apply in particular cases even without an NDA.
Data protection law won't apply unless the company name effectively identifies an individual (e.g. one-person companies).
Legal issues aside, I do think it is good manners to ask customers for permission to use (especially to publish) their names. If such use is problem for them, then you could damage customer relationships. If it isn't, then why not ask for permission? In my experience, SMEs are usually happy to be mentioned.
NB using logos gives rise to additional issues, especially in relation to copyright law.