Your daughter sounds a lot like my older son. He's also four, and he's also very much attached to us. He won't do things that he considers fun - like gymnastics class, soccer, etc. - unless mommy or daddy is participating with him (not even hanging out nearby, we have to be next to him). He's a very independent child in every other way - and if you get him away from us, he acts extremely independently, like at pre-school.
I figure it comes from a few things. One is insecurity and abandonment issues with new things - as many children are. He does much better when he's familiar with the environment and familiar with the person. As such, we try to put him in positions where he will succeed. We give him a chance to meet teachers/etc. ahead of time. We take him to places before he's asked to separate from us. We spend a great deal of time prepping him for anything that will be significant. This works fairly well; but, he still has a day or two of significant difficulty as he adjusts to new things (30-60 minutes of difficulty). Us coming to pick him up on time every day is of course also a major plus.
The second issue is likely that he wants to have a lot of attention, and when we're not around he doesn't get it as much (of course!). He knows this, so he resists being put in situations where he won't get that attention. This has become less of a big deal as he ages - and we have a second younger child, so this also has undoubtedly helped and hurt both, but on balance helped.
The attention issue is more complicated to work with, but we've found that simply putting him in more situations where he doesn't have the high degree of attention - whether it be preschool, gymnastics class, etc. - seems to help to some degree. Giving him plenty of attention when he is around us is also good, in my opinion, as it helps keep him assured that he will get our attention when we can give it - but giving him lots of opportunities to learn that he can be successful without our attention is needed.
Letting him know that he can do things on his own, but not pushing too hard, has worked quite well there. For example, for the last six months or so, I've suggested that when he needs to go to the bathroom in a restaurant, and it's visible from the table, that he simply go. This was a silly suggestion six months ago, but I never pushed hard; simply letting him do it if he wants to. In the last few weeks, he's started to be willing to - not every time, but still, sometimes, particularly when it's in his interest to do so (like, when I was carrying his sleeping brother and couldn't leave the table last week).
In your particular situation, I would suggest a bit more familiarization with the uncle if that is a possible issue (if this isn't someone she already spends a reasonably large amount of time with), prepping her for the situation (letting her know all of the details a few different times in advance), and then... going. She may cry, she may have a hard time, but she'll also learn. Perhaps a few shows that discuss this sort of issue - there is a Daniel Tiger episode that's excellent, with the repeated song/phrase "Grown ups always come back", for example; many other shows cover similar issues.
You might also talk to the uncle, and make sure he is engaging her. A four year old does need engagement, perhaps not constantly, but periodically. Find something fun they can do together - maybe a game (there are lots of games that I find fun to play that my kids do also; games like Spot It or Memory for example, which is at least mildly challenging for the adult. Or if the uncle has a particular interest, perhaps he could teach her about that interest - anything is interesting to a four year old, be it historical interests, woodworking, fixing cars, whatever he is interested in.