As of Outcasts & Remnants v1.30 and Project Valkyrie v1.10, Sarah Lyons can be romanced. To romance her, build affinity by doing the following:
 
1) Complete the quest “Wag the Dog”, then talk to Sarah. Choosing the "Flirt" option is your first opportunity to hint to Sarah your romantic interest in her.

2) Complete the quest “Break the Wheel”, where you help Sarah fight off the BoS counter strike.

3) Complete at least up through Carter’s mission in “The Revolution will not be Televised” where you overthrow Institute management. Then talk to Sarah. When she opens up about her feelings, you'll have a second opportunity to flirt.

4) Finish "The Revolution will not be Televised”, then exit the reformed Institute and talk to Sarah, at which point she will proposition you.

5) Accept her advances, and you'll head back to her place. If you reject her initially (by saying just friends or that you don’t find her attractive), you won’t get a second chance to romance her later on. She’ll never bring up the topic again, and you will be permanently in the friend-zone.

6) For additional romantic encounters beyond the first one, you need to do side quests for Sarah by asking her for work. Each time you complete a side quest with her, you earn affinity. Each time you sleep with her, affinity is used up. If you’re above the affinity threshold, the “Your Thoughts” dialogue menu option will instead read “Romance Sarah”. Using that option will initiate the romance dialogue and scene.

7) If you reject Sarah on a follow up romantic encounter, it will merely reduce affinity, without preventing future encounters, and you’ll generally have to do another side quest for her to get the romance dialogue option again.
 
If you’ve already completed “Wag the Dog” and “The Revolution will not be Televised” in your current play-through, complete “Break the Wheel”, then Sarah will proposition you.
 
After romancing Sarah, her greetings will become friendlier, her thoughts will become more flirtatious, and she’ll occasionally make suggestive idle comments. The Sarah romance scenes transport you back to her place. The dialogue in the pre- and post-romance scenes is randomized with a dozen different combos.

After sleeping with Sarah, she’ll comment on the encounter, then head to the shower for approximately 60 seconds, then get dressed. Before hitting the showers she'll also run a load of laundry. Check the washer, dryer, and dressers each time for new outfits + accessories. After showering, Sarah will put on one of a dozen outfits. If you don't like the outfit she puts on, just ask her to trade and wear something else.

The Sarah romance dialogue uses explicit language. If you have nude mods installed, Sarah will be naked. If not, she'll wear underwear. The sex scene is fade-to-black, like the Magnolia one in the vanilla game. There are no sex animations included in the mod. If someones wants to make a patch with actual animations on the site which shall not be named, go for it.

After romance, you’ll gain a temporary stat buff called Lyon’s Pride (strength, damage resistance, and XP bonus) which lasts for 12 in-game hours, or 36 minutes in real-world time. If playing strategically, save up the romance points and sleep with Sarah right before heading into a tough battle area.

The romance mechanism can also be used as a means of de facto fast travel from any interior to Sarah's bunker, and you can also fast travel out from inside Sarah's bunker.

For better immersion, holster your weapon before engaging in the romance dialogue with Sarah. The spawn point for the romance scene is the living room, but if you have lots of followers, Sarah may spawn in the bedroom or bathroom instead.
 
Sarah's romance doesn't affect the Lover's Embrace perk. If you sleep in a settlement bed, Sarah will ignore it, and you can continue sleeping with your other companions and earning the Lover’s Embrace perk as usual.

For Valkyrie and/or Fiona to have romance options in a future update, there need to be ZERO instances of stalker behavior from the community. That starts by not posting the Sarah romance scenes on YouTube or Twitch. There's one romance play-through video on YouTube, if you get stuck, and that should be it. Please be cool so as not to screw it up for everyone else.

If you're a YouTuber or Twitch streamer and that's not clear, please see the mod permissions section and the bottom of the "Random Questions & Answers" article. TL;DR = to have voice-acted mods, we can't have your viewers harassing voice actors. If you don't post the video, the trolls and stalkers from your audience won't see it, and the rest of us can produce and play mods in peace. The arrangement whereby you monetize videos based on the work others do for free, while a small portion of your psychotic viewers proceed to harass/stalk/issue death threats against the people providing you with free content, is untenable. This far, no further.