How I should deal with this situation?
This is an open-ended question, which means any answer is equally acceptable (not equally liked, mind you, but equally acceptable.)
You can simply ask: "I'm curious why you call her Harry instead of Harriet. Can you explain that to me?" In most cases, it's a sign of affection for the child and is not a reflection on you.
However, I think there's an important element here that I would encourage you to examine carefully: your own feelings about yourself.
My partner is fine with Harriet being called Harry, but I feel sad and frustrated that the name we chose is not being used. I am not sure the reason behind it, but it feels to me as though they have decided that the name we chose was not acceptable. I have had a number of sleepless nights because of this and when her family is using the name to my daughter I feel very tense and it makes me feel down.
Your partner is not supporting you (no judgement, just a fact.) That probably causes you to feel unsupported and/or alone. You use the words "I feel" a lot: I feel: sad, frustrated, tense, down, [judged and found wanting]. I say judged because "it feels to me as though they have decided that the name we chose was not acceptable" isn't a feeling but a belief or notion that this reflects poorly on you (they are judging you.) I say, your feelings about yourself because hurt feelings rarely occur in a vacuum. If, for example, you feel, say, "less than", unrelated things might reinforce that feeling of being "less than". If you feel, say, alone in this world, unrelated things might reinforce that feeling of "aloneness". Our feelings about things are colored by our experiences, which in turn affect how we feel about ourselves. So it's important to work on and understand them.
It's not necessary in Parenting.SE questions to go into how in-laws treat you, and you haven't, but if, in general, they treat you well and with respect, then please assume they feel no differently about you regarding the name you chose for your daughter. Changing a child's name makes it more personal to the person doing it: a long ago President's son, John Fitzgerald Kennedy Jr., was affectionately called "Jon-Jon" by his family and a nation. A child named Hannah will be "Hannah Banana". If, on the other hand, they like and respect you, but don't like the name you chose for your daughter, that's just tough beans for them; it's still your daughter's official name.
If, however, your in-laws tend to treat you less-than-well, this is a situation that reinforces the notion that they disapprove of you, and anything doing that will be hurtful. It will hurt that your wife basically agrees with them and not you, and it will remind you every time you hear that name that you are not loved/respected as you (and all people) want to be loved/respected.
In either case, if it's causing you a loss of sleep, you need help to sort out your feelings ("I am not sure the reason behind it...") and may need someone to help you do that, someone who can also challenge your belief system without you feeling threatened. That person is usually a stranger, like a therapist or an advice columnist, or someone on the internet.
My partner and I had always thought that if a short name were to be used it would probably be Harry, but I didn’t anticipate it might be used so soon.
Ok, so it happened sooner than you expected. You were unprepared for the actions of others. This happens to people every single day, because we can't control others, only ourselves. Can you shrug it off? If not, why?
If you really want to be active about their choice of names for your daughter, you can remind them every time they are about to see her, upon opening the door, "Hello, X, and please remember that my daughter's name is Harriet, not Harry." That, however, might signal that you don't respect the feelings of others about her name, or them in general. Sometimes it's OK to put your feelings first; just remember that everyone has them, and you can't control what others think or feel.
