People are being too prescriptive on this thread.
OP, you can call her what you wish to.
There is a difference between shortenings and nicknames.
Shortenings are usually shortened versions of longer names. Often in the UK these are things like Tom for Thomas and Eve for Evelyn. Or they can come from different parts of the word. Beth or Liz for Elizabeth. Izzy or Bella for Isobel.
But there are also traditional shortenings that are a bit more wide-ranging and where the letters change. Jack for John for example. Bill for William. Molly and Polly for Mary. Daisy for Margaret! This is mainly because so many people were called John or William, say, in the same family that they needed ways of differentiating. So Molly from Mary and Polly when there are too many Mollies. Or Will from William that changes to Bill.
There is also a common thing of using a second name as the day to day name, also due to perhaps having a parent with the same name or too many people with the same name at that time.
So all these traditions exist and it really creates quite a wide spectrum of people being used to stuff.
Evangelia is actually a great name for options as there are so many sounds in there.
Ev, Evvie and Evie and Eve
Van, Vannie and Vannia
Anne and Annie
Angie and Angeli
Geli and Gia
Ellie and Elia
Lia
But even added to that things like Lila work fine. You just call her that day to day and have Evangelia to be used if she wants for full. I don't think anyone in real life would turn a hair despite what a mumsnet name purist might say! People on here are always suggesting CeCe for Cecily or Cecelia for example. Which aren't the same sounds and always seems like a bit of a jump to me. So it's just down to familiarity isn't it? If you can have CeCe for Cecilia, why not Vivi for Evangelia? No reason at all.
People going on about not be able to control nicknames are being over-controlling if you ask me. Yes a person grows up and calls themselves what they like, but that's fine. Even so, if someone intros themselves as Lila people are very unlikely to suddenly call them Eve, unless the person themselves wants to be called Eve, and if they do, so what, you can't control that anyway. She can call herself anything with her friends as anyone can.
If you call someone William, Will for short, they are unlikely to suddenly be called Liam. They can be if they want but they'd probably have to make an effort to do this. If someone is Liam, they are unlikely to suddenly be Will. If someone is Elizabeth known as Lizzie, they are unlikely to be Beth from their friends unless they want to be. If they are always known as Beth, it's unlikely their friends will suddenly switch to Lizzie, unless the person wants them to. It's different perhaps with shorter names. Stephen goes to Steve and people will naturally say this because it's a short name with no other obvious shortenings. Thomas to Tom. But Evangelia is so long, it has a lot of options.
So I wouldn't worry so much. If you like Lila, go for that.
The shortenings I like best are Lia (because it's beautiful) and Van and Vanna or Vannia (because this sounds different and cool to my ears and would be great for a teen!). Evie and Ellie are very usual at the moment so I might go for something different as they don't sound so exciting to me, and if I had an amazing name like Evangelia I might want a slightly more interesting shortening.
The only thing about Lila as you found out is people pronounce things all sorts of different ways. But as a shortening of Evangelia it seems fine to me. Go with what you love, OP.