We may face some issues with the timing of Ubuntu 20.04 and its effects
If everything goes as listed in Ubuntu's 20.04 release schedule, Ubuntu 20.04 LTS will be released April 23rd, and this release will start a roughly one year countdown on security updates for Ubuntu 16.04, which would stop a year later in April of 2021. Under normal circumstances, we would spend a chunk of May bringing up 20.04 in our general Ubuntu install system, deploy some initial systems for testing, and then ideally start targeting an August upgrade of our user-facing 18.04 machines (which we try to keep on the current version of Ubuntu). We would also start working through our decent number of 16.04 machines in order to replace them well in advance of the support deadline (we have both 16.04 and 18.04 machines because of how we handle Ubuntu LTS versions).
These are not normal times. It seems unlikely that we'll be back in our office before July (cf), and while we can do some amount of work on 20.04 remotely (especially now that I can run VMWare Workstation from home), this only goes so far. We need to test 20.04 on real hardware before we deploy it, and we definitely can't remotely upgrade existing servers to 20.04. Also, we're unlikely to be able to immediately launch into new things the moment we're back in the office; there will probably be a backlog of problems to be dealt with first.
If everything goes back to normal around the start of July, we can almost certainly still squeeze everything in. We may have to push 20.04 upgrades for some machines to inconvenient times (for users and for us). In particular, we don't have that many 16.04 machines and so I don't think we'd have any real problem meeting the end of support deadline for it. However, that's the optimistic case. A more extended period out of the office would start to push at the 16.04 deadline, with us potentially looking at upgrading multiple critical machines (such as all of our mail servers) within the span of a few months at most (while we're dealing with many other issues).
If there is a general extended period of disruption all over, it's possible that Canonical will take the unusual step of extending free Ubuntu 16.04 support before the normal one year time after 20.04 is released. Apart from that, Canonical has already said that Extended Security Maintenance will be available for 16.04; we may be forced to buy that for crucial and hard to migrate machines that we can't get to in time, at least as a bridge to buy us extra time.
(Hopefully we could arrange something lower than Canonical's standard rates, which are rather expensive for a university department.)