I don't have an end-to-end solution, but I will say as a former Pebble user that I'm delighted with my BangleJS watch. It's inexpensive, I charge it once a week, and it has the usual activity tracker hardware including heart rate monitor. There are a ton of apps. It's less "hacker" and more polished for non-technical users than you might expect; if you're running Linux, this is absolutely within your comfort zone. It also has a temp monitor, so it can detect when the watch it being worn or not.

The customized GadgetBridge Android app is good, most of the programming is in JavaScript, and the developer community is active and responsive.

A quick glance through the app "store" (they're nearly all free) shows there's a heart alarm app, which seems aimed at notifying the wearer. Another app publishes heart rate over bluetooth, so someone must have a pair device that monitors heart rates.

I didn't see anything obvious in the ecosystem, but since the data is going to GadgetBridge already, it's most of the way there; if GadgetBridge could publish to web hooks, with ntfy you could get alerts from watch to partner's phone. I haven't yet looked through the GB issues to see if someone has already requested it, and I've only done light digging in the app itself; it may already support this.

It's worth investigating.

Streamers sometimes use a heart rate monitor widget on stream. There might be multiple solutions here but checking out pulsoid.net it seems like a free solution can be made here if it is compatible with the smart watch used. Android app to webpage might be possible with setting up a local host server. Don't know what it'll do to the battery life of your watch and if the widget works without OBS integration. Maybe someone familiar with this can weigh in?

If you have an iPhone, you can get an Apple Watch and set it up using the other person’s phone. The health app will tell you what the person’s heart rate is. It will also do sleep tracking and check blood oxygen levels. It will also notify you if the person falls. It can also be used to contact 911 if it is in range of your phone, or if you buy the cellular version and add it to your cell phone plan.

If you live near an Apple store, stop in and ask about it. The trick will be to set up/link your spouses watch with your iPhone so you can continuously monitor the other person.

I remember seeing this ring built specifically for this purpose. It was an unobtrusive metal band and you could synch it up to its pair and it would squeeze your finger in time with the pair's heartbeat sensor, I'll see if I can find it again

I don't believe this is the one I came across but was all I could find with a quick search. The website seems a li'l sketchy based on grammatical errors so I'd do some double checking into it but it seems like what you're looking for. https://my.thetouchx.com/hbring/howto-use-hbring

I'm afraid I can't offer any recommendations but I just wanted to say how beautiful it is to think of you keeping track of each others heartbeats while you're apart. That's love

My baby monitor nanit has a special blanket you can put on the baby to track breathing and if there's a problem you phone app will alert you. You may have to pay a subscription service to use this feature.

https://www.nanit.com/products/nanit-breathing-band?srsltid=AfmBOoocMwOc__rbAmCl5P1wqM-yK_Y9xhQ3HOPlHekpmaDD1nDrl4hg

Not sure how well it works on adults of course but worth a shot.