Term for a syndrome where someone falls in love with their rescuer?

What about referring to “Prince Charming” i.e. the archetypal handsome rescuer who saves the damsel in distress, and whom she always falls in love with, and then lives with happily ever after.

Prince Charming is a fairy tale character who comes to the rescue of a damsel in distress and must engage in a quest to liberate her from an evil spell. This classification suits most heroes of a number of traditional folk tales, including "Snow White", "Sleeping Beauty", and "Cinderella", even if in the original story they were given another name, or no name at all. Often handsome and romantic, these characters are essentially interchangeable, serving as a foil to the heroine; in many variants, they can be viewed as a metaphor for a reward the heroine achieves for the decisions she makes. [...] — https://en.m.wikipedia.org/wiki/Prince_Charming

However, the term Prince Charming Syndrome, is already used for someone waiting (in vain) for the perfect relationship :

Dr. Robi: The person who has the “Prince Charming Syndrome” is also someone who is very inflexible when it comes to love and relationships. They have such a romanticized version of what love should look like [...] - https://m.huffpost.com/us/entry/us_6498436

but perhaps you could say a “Prince Charming” relationship or “Prince Charming & damsel-in-distress” relationship or suchlike ?

There are the terms "white knight" and "knight in shining armour".

white knight

n. 1. One that comes to the rescue; a savior.
American Heritage Dictionary

n. 1. a hero who comes to the rescue.
Random House Kernerman Webster's College Dictionary

1A person or thing that comes to someone's aid.
Oxford Living Dictionaries

The terms are not really used (for women). The term may have originated from the knight errant stock character or the "damsel in distress" narrative. Note the "white knight" disambiguation page on Wikipedia lists "knight errant" as a possible meaning. The article itself has a section on romance, and also mentions the term "damsel in distress".

Isn't there a white knight upon a fiery steed? (song lyrics)
I need a hero

Knight in shining armour

knight in shining armor
A selfless, chivalrous man who helps a woman in distress.
When the police officer pulled over to help the old woman change her flat tire, she hugged him and said he was her knight in shining armor.
Farlex Dictionary of Idioms

As to using either of these terms with the word "syndrome", you'll want to be careful. The term "white knight" has acquired a generally pejorative meaning of a man who is very obliging towards woman, affects this behaviour, or stands up for women's rights. This is the general meaning given in Urban Dictionary. It also lists "white knight syndrome" from 2005, there are various meanings given, including when a man becomes attracted to a damsel in distress. We can't know what currency these definitions have, so I would be careful.

Following on from my caution above, when using some of these relatively uncommon terms you're quite liable to being misunderstood. For example if we take your example of Florence Nightingale Effect or Syndrome some sources will define it as either a caretaker developing romantic feelings for their patient, or the reverse, or both. Segen's Medical Dictionary also gives an entirely different meaning.

Both the terms I mentioned are idioms for a person who saves or rescues another person (nearly always man saving a woman). As far as I know there is no well accepted term for the phenomenon you talk about. But these may be of use to you if you want to use your own term.

It’s called “transference” where one begins to attaché romantic feelings to a person in a professional protector role such as a doctor, therapist, policeman and such.