Curing meats without nitrite is totally possible through the proper synergy of lowered pH, salt concentration, and reduced water activity, so I'm thinking your small format sausages are probably working out because you are getting the appropriate log reduction in pH quickly enough, and an eventually low enough water activity to exclude botulism. Traditional prosciutto is salt buried, which is another way to exclude botulism because it cannot survive salt concentrations above 5%. I advocate for the use of nitrites (either natural, via celery juice extract, or synthetic) in curing because on the home scale I cannot guarantee that users of my books are going to achieve the proper parameters for safe curing without them. But it IS entirely possible, as your sausages have proven. The salt content you are using is likely not the factor, as the other user mentioned, but the pH and dehydration over time are such that botulism can't thrive. As for how nitrite is produced, it is either synthesized in labs or mined in the form of nitratine. You can make it from urea, as the other user mentioned. You can also make it from other chemical reactions but you might not want to be dealing with some of the chemicals involved (aluminum nitrate or lead nitrate, for example).
The only issue with celery products are that their nitrite composition is inconsistent, meaning that in a specific lot of that product you could have higher amounts of nitrite than are needed to safely cure the product, or you could have lower amounts. As such, package recommendations are for the use of a slight excess of celery product, which will definitely get you to a safe end product, but potentially (though not absolutely) with an overall higher actual amount of nitrite.
The compound nitrite is naturally occurring and we consume it regularly when we eat vegetables. Normal metabolism converts it to nitric oxide-- no problem. Same as getting a little sun on your back. Similarly, in cured meats that are NEVER cooked, such as your sausages, the nitrite is converted by microbes into nitric oxide, with trace secondary metabolites of nitrosamine. Nitrosamine is a known carcinogen, however the amounts produced in cured meats that are not cooked have not been shown harmful. The issue with nitrosamine arises when humans cure meat with nitrites and then cook it, producing nitrosamine at high levels. This is the rub with store-bought cured meats, because the FDA doesn't allow commercial sale of cooked, ready to eat meat products that are produced without nitrites. Whether the producer uses celery products or synthetic nitrite, the end result is the same: nitrosamine.
I urge my students to take control of their food by using what science they have access to to inform their decisions. In the case of bacon, for example, the product will cure relatively quickly and be cooked at temperatures well above boiling point for longer than 20 minutes-- a process which is known to kill botulism toxin. (note that botulism toxin is a secretion of the spores of the botulinum bacteria. The bacterium itself will not make you ill-- it has to be reproducing to make spores and those spores secreting toxins to produce the illness). In the case of this hot-smoked bacon, the use of nitrite is indeed optional. If you were going to COLD smoke the bacon, however with temperatures not rising above 80F/26C, then I would recommend use of a nitrite.
Always thinking whether the product will be cooked or not. If not- always use nitrite OR ensure through pH readings and water activity readings that you are safe. If you are cooking, are temperatures above boiling point and cooking times longer than 20 minutes? If so nitrite may be omitted. You will see grayer coloring in products that do not have nitrite in them.
By the way, parameters for botulinum: It generally cannot thrive in water activity below 0.97 and pH below 4.2. Safe curing of sausages require a reduction in pH below 5 within the first two days of fermentation and an eventual pH of 4.2 or less if you are trying not to use nitrites. Hope this helps!