Interesting question! Unfortunately cycle.travel’s route-planner isn’t really set up for offering options for different types of route: essentially, adding more options means buying more servers, which is prohibitively expensive (more here). But if there’s a particular road where you think cycle.travel is making the wrong choice and could choose a nearby road with a cycle lane instead, I’d be very interested to know.

Do cycle lanes get weighted by c.t's planner by width? Do cycleways? (In other words, is it worth tagging them in OSM with their widths?)

Do cycle lanes get weighted the same globally? I'd venture that a cycle lane in NL is on average more likely worth using than one in GB.

Width – no, not at present. Width data is very patchy in OSM, and in some cases in the UK actively erratic: a bunch of DfT cycle lane data was brought into OSM years ago, including a lot of (in retrospect) rather inaccurate widths.

cycle.travel does differentiate between OSM cycleways (highway=cycleway) and footways-on-which-bikes-are-allowed (highway=footway, bicycle=yes), which is often a proxy for width; and, of course, it prefers off-carriageway cycleways to on-carriageway cycle lanes.

Weightings do differ from country to country. (And from county to county in the US, but that’s another story!) The Netherlands is quite an interesting example. Unlike many other countries, cycle.travel doesn’t have motor traffic data for the Netherlands, but in practice it doesn’t matter: the infrastructure is so copious that the algorithm is unlikely to send you on a busy road anyway.