Hello and welcome. It's lovely to hear she's a happy hamster and no behaviour issues. Looking forward to hearing more about her! She will probably be fine for now but as Elusive said, it would be good to upgrade when you can. She has made her own deep bedding in one corner by the sound of it
It's always difficult to achieve deeper bedding in barred cages, due to the limitations of cage base size, although there are ways round that, if a cage is tall enough, as you can put cardboard or grass or hemp mats round the inside of the bars (or perspex but that is more expensive).
When switching from shavings to paper based, it is tricky. Assume she's only about 12 weeks old? Although if you've had her for six weeks she should be well settled into her cage by now. When I first switched from wood shavings to paper based, I did it gradually, I replaced half the bedding with paper based and mixed it all in together so it still smelled familiar. Then gradually, when spot cleaning, or adding more if it went flat, I just carried on adding paper bedding. Eventually it was mostly paper and very little shavings. It did take quite a while for the last of the shavings to disappear as I didn't want to do a full cage clean and replace all of it. I think that would be perfectly fine, providing the current shavings are kiln dried pine and dust extracted. If it just says "softwood" it could be a mix of all sorts and not just pine.
On the other hand, some people just replace it all but don't clean anything else in the cage at the same time. This is a bit unsettling for the hamster, if removing all their familiar scent though. You could do something in between
Replace half for now and mix it in, then keep spot cleaning and adding more paper that way, then after a few months, when you do a substrate change, add all new paper but keep back some of the existing paper bedding (picking the shavings out as much as possible) and spread that on top so it still smells familiar. That top layer will eventually get spot cleaned out and replaced anyway.
The Criceti 15 is a bit on the small side. I think with the shelf and new wheel there won't be a lot more space for adding much, and if you fill the base with substrate it would still need to be low at the wheel area so the wheel fits in the 39cm height.
Something like the Criceti 100 or the Savic Plaza would be a much better option. They have a bit of extra height and are just 20cm longer but it gives a lot more floor space. With the extra height you can fill the base with bedding - or go up to 8" deep with something round the inside of the bars, and still have space for a 12" wheel above that. I've used the Savic Plaza for Syrians for a few years and it works well (and has a nice big front opening door which makes access easy and easy interaction with the hamster) And you can attach a wheel to the bars higher up so it sits above the substrate.
I would have a look around for a second hand one, as it's been out a few years now and there are quite a number of them available used for between £30 and £50. Places like ebay and Gumtree etc.
Some people make diy cages if you like that idea and the idea of a tank style cage, but a used Savic Plaza is probably the cheapest option.