In general (even after getting hold of a PC), I always maintained a simple log book next to the phone in the workshop. Anyone taking the call (or receiving a visitor - nurse or whomever - at our basement lair) would be required to jot down a few entries in the appropriate columns (date, time, department, brief "problem" and what-have-you). Naturally, the Chief would keep on top of those entries (what is now often referred to as "follow up").
My only experience of a "call logging centre" per se was at a large Desert Hospital many moons ago. I worked for an O&M contractor, and our client (the MOH) had a very "active" Supervisory Team on site. They decided they wanted a "single point" for problems to be reported.
Fair enough (at least at first glance), and the "call centre" was duly set up. It was a small office, centrally located off a main corridor, with single internal phone extension. Users could also drop off "complaints" (not to mention sweet papers) in a letter box at the door. Although the client seemed happy enough, as far as we (the biomeds) were concerned, the whole thing was a failure (not unexpectedly, I must admit). Why? Because the "engineer" assigned to the office (which was only manned 9-5, if I remember correctly) was unsuited to the task. For example, the guy seemed continually confused about which department (property control, facilities, security, biomed etc.) should respond to the Request for Service ... resulting in (sometimes) long delays in messages arriving where help could be provided. Requests got "lost" (probably in the "too difficult" bin) ... and many more "issues", all of which added up to a pretty poor show from the "customer satisfaction" point of view.
In almost all cases, by the time the message reached us from the "call centre" - be it by phone or via an internal messenger - the problem had already been addressed by us; often hours before.
Meanwhile, I simply continued to encourage the nurses, lab techs and radiographers et al to carry on phoning the workshop when they needed help ... or simply catching me as usual as I did my daily (sometimes twice daily) tour of likely departments. Or, if desperate, by paging me (remember those things?).
I would urge all biomeds to do the same:- cultivate the trust of your "customers" (and don't let them down). And avoid having any "middle men" - such as the "call centre", Estates (Facilities) department (or any other interloper) between you and your users. Those were simpler times (in simpler places) perhaps, but the mantra of the "biomeds are your friends" is one that I liked to cultivate.