Like the title says, I have a very stable machine running 16.04 at the moment and I'm thinking of updating it to 18.04 (and then maybe 20.04 after). I will want to go the update route rather than a clean install so will I be able to choose 18.04 from Software Updater up until 16.04's EOL?
Yes. Later LTS releases reach End of Standard Support after earlier LTS releases.1 If you have an LTS release that is currently supported, there should be no problems upgrading it at any time while it is still supported. (You can could even attempt to upgrade it after it is no longer supported, which usually succeeds but is sometimes more difficult. But I recommend against needlessly placing yourself in such a situation.)
Regarding the specific releases you mentioned, as shown on the releases wiki page, Ubuntu 18.04 LTS reaches End of Standard Support in April 2023, which is after April 2021. You should upgrade your 16.04 LTS system before it reaches End of Standard Support in April 2021, mainly because if you don't then, at least for a short time, you'll be running a release that doesn't receive security updates, which is undesirable. But there is no need to upgrade it before that unless you want the new features of a newer release sooner, or need a bugfix that is is provided for 18.04 LTS and not backported to 16.04 LTS.
Of course it is always possible, when upgrading from one release to another, that something could go wrong. But there's no reason to think this is more likely if you wait until the earlier release has almost, but not quite, reached End of Standard Support. For example, updates for security and other stability bugs could introduce a bug that complicates upgrading, but they could also fix a bug that complicates upgrading.
(Note that if you end up deciding you want to move from 16.04 LTS to 20.04 LTS, you should still upgrade to 18.04 LTS first and then to 20.04 LTS. Usually this is what happens automatically if you perform two release upgrades starting from an LTS release.)
1 There was a time when this was not fully true. 8.04 LTS and 10.04 LTS had unusual (and confusing) support periods, with a very small number of packages supported for a very long time. However, this is fully true for all releases that are currently supported or have been supported in the last few years, and it is likely to remain true many years into the future. Even if it does not, the specific releases you're talking about already exist and their support lifecycle has already been determined. It's unlikely that the support lifecycle of an existing release will be extended beyond what has been planned, though you can get limited support for an LTS release beyond its End of Standard Support through the Canonical Advantage program.
Yes, technically it'll still work after it's EOL (end-of-life) too.
For non-LTS releases that period can be rather short (17.04 was days having no meltdown patches), some release repositories get moved in weeks, others months. Officially it's only listed as occurring after the EOL, without detail of how how long after.
Because 16.04 LTS (long-term-support) has extended support (ie. ESM) the extra time is longer than months... (but you shouldn't wait for security reasons, unless you're choosing to use extended/ESM support. See https://wiki.ubuntu.com/BionicBeaver/ReleaseNotes
Example. I use a mirror (https://launchpad.net/ubuntu/+archivemirrors) provided by my ISP, it still provides packages for precise (12.04) & trusty (14.04) because those releases have extended support. Xenial (16.04) will fall into this category. You'll find this in the main archive too, however mirrors are free to not provide ESM support so it may have been dropped anytime after EOL.
Also be aware many official teams erase their PPA (3rd party software sources) after a release has gone EOL, so you cannot really make any assumptions (this is up to every team, it's often done when people think of it, occasionally forgotten, and some PPAs are never emptied).
If you need details on extended support, please see https://ubuntu.com/esm (and note a number of without-$cost options are available, 50 machines for Ubuntu members like myself)
It's best you don't make assumptions, Ubuntu Studio 18.04 was not a LTS or long-term-support release which was clear in it's release notes, and subsequent notices about getting extended support (that extended support to 3 years same as other Ubuntu LTS flavors). So read the release notes, which include details on upgrading from prior releases (I already provided the link for 18.04 LTS which contains instructions for "Upgrading from Ubuntu 16.04 LTS or 17.10"). Any problems detected, and rules the teams work by will be provided in those notes, and they can change over time so read the specific notes for your release (https://wiki.ubuntu.com/XenialXerus/ReleaseNotes for 16.04)