Nightingale

The first thing to point out is that Nightingale, unlike every other Berserker, is first and foremost a support servant, not a damage dealer. It's a bit contradictory, but she's not meant to actually do all that much damage, instead letting other servants do the damage for her. Honestly, she'll put in work for you once you get used to her (I actually rolled her myself on GSSR), namely figuring out how to keep a berserker alive for any stretch of time.

Her heal is strong on a reasonable cooldown, her second skills means she can have some damage against servants, but more importantly will give her defense against almost every servant and many generic mobs in the game. However, it's her third skill that you'll want to hear about.

Every servant you listed off has a buster Noble Phantasm, and having a 50% buster buff for three turns is a colossal boost in damage, and with a five turn cooldown, it's very nearly gamebreaking.

For Illya, she's great for another reason. In addition to buffing her Buster Noble Phantasm damage with Angel's Outcry, Nightingale's Noble Phantasm provides a party heal and removes debuffs. Illya's Noble Phantasm gives her two debuffs whenever you use it, so if you chained Illya's into Nightingale, Illya wouldn't have to worry about the attack and defense demerits on her NP.

Overall, Nightingale is alot harder to use then your standard 'Click-buster-and-win' Berserkers, but she's really strong at higher levels with the right team set up.

I pulled Nightingale as my very first 5* servant, and haven't had many regrets since starting 3-4 months ago.
She does surprisingly well in two different team compositions, the first is in a strong arts team (waver, tamamo, etc.) as when she eventually gets her interlude, the -50% attack debuff can cripple an enemy team's NP completely, and paired with Waver's stun, you can stall out enemies quite hard, even with the inherent berserker squishiness. Granted, this team setup takes team, and some very premium SSR units, but if you're willing to try it, its actually pretty fun, as she deals decent damage, and the continual healing and NP spamming makes this art team surprisingly hard to take out.

The second, and way more popular team setup is slapping her onto a berserker centric team. I paired her up with Ibaraki Douji, Hercules and Frankenstein/ Tamamo Cat. Against bosses, a +50% buster damage boost with a buster brave chain starting with a NP by Hercules (or any servant with buster NP and 2 buster cards) can easily take out half a boss' health no problem (this very simple setup carried me the Entire way through each and every single Camelot boss, even the insane ones like Ozy, Gawain and Artoria Lancer)

In general, she's fantastic as a backup strong heal (1st skill, or even NP) and when supporting a berserker, can annihilate very many encounters. In your case, when you build a team, bring out Herc as well, and watch as all enemies die immediately by Herc's hands (with some very nice help from Nightingale).