Basic functionality can be obtained with a webhook:
1/ updown.io > Settings > Webhooks > Add "https://ntfy.sh/TOPIC-NAME?priority=urgent" > Save
2/ updown.io > Settings > Send test notification > Select the webhook > Send

I initially missed it in the ntfy.sh documentation, but the headers can be replaced by URL arguments
- https://docs.ntfy.sh/publish/#webhooks-publish-via-get
- https://docs.ntfy.sh/publish/#list-of-all-parameters

Full support (title, etc.) would be nicer. Maybe it can be easily implemented with a pre-defined template:
- https://docs.ntfy.sh/publish/#pre-defined-templates

Note: the urgent/max/5 priority basically transforms the ntfy.sh mobile app into an oncall pager, overriding do-not-disturb settings. This works best when using push notifications (Google's FCM or Apple's APN). For that, install the Play Store version.

Building on Mihai's suggestions, here's an example webhook URL using an inline template:
https://ntfy.sh/mytopic?auth=ACCESS_TOKEN_FROM_NTFY_SETTINGS&tpl=yes&t=%7B%7Bwith%20index%20.%200%7D%7D%7B%7Bif%20eq%20.event%20%22check.down%22%7D%7DDOWN%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%20if%20eq%20.event%20%22check.up%22%7D%7DUP%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%20if%20eq%20.event%20%22check.ssl_expiration%22%7D%7DTLS%20expires%20in%20%7B%7B.ssl.days_before_expiration%7D%7Dd%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%20if%20eq%20.event%20%22check.ssl_invalid%22%7D%7DTLS%20invalid%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%20if%20eq%20.event%20%22check.ssl_valid%22%7D%7DTLS%20valid%20again%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%20if%20eq%20.event%20%22check.ssl_renewed%22%7D%7DTLS%20renewed%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%20if%20eq%20.event%20%22check.performance_drop%22%7D%7DApdex%20-%7B%7B.apdex_dropped%7D%7D%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Belse%7D%7D%7B%7B.event%7D%7D%3A%20%7B%7B.check.alias%20%7C%20default%20.check.url%7D%7D%7B%7Bend%7D%7D%7B%7Bend%7D%7D&m=%7B%7Bwith%20index%20.%200%7D%7D%7B%7B.description%7D%7D%5Cn%5Cn%7B%7B.check.url%7D%7D%20%28%7B%7B.check.token%7D%7D%29%20%C2%B7%20uptime%20%7B%7B.check.uptime%7D%7D%25%7B%7Bwith%20.downtime%7D%7D%7B%7Bif%20.duration%7D%7D%5CnDown%20for%20%7B%7Bprintf%20%22%25.0f%22%20.duration%20%7C%20duration%7D%7D%7B%7Bend%7D%7D%5Cn%7B%7B.details_url%7D%7D%7B%7Bend%7D%7D%7B%7Bend%7D%7D&p=%7B%7Bwith%20index%20.%200%7D%7D%7B%7Bif%20or%20%28eq%20.event%20%22check.ssl_expiration%22%29%20%28eq%20.event%20%22check.ssl_renewed%22%29%7D%7Dlow%7B%7Belse%20if%20or%20%28eq%20.event%20%22check.up%22%29%20%28eq%20.event%20%22check.ssl_valid%22%29%20%28eq%20.event%20%22check.performance_drop%22%29%7D%7Ddefault%7B%7Belse%7D%7Dhigh%7B%7Bend%7D%7D%7B%7Bend%7D%7D