Does swallowing salty saliva after performing ablution with salty water invalidate the fast, given that no tissue was available to spit it out? And what is the ruling on this issue?

Your fasting and prayers are valid. Swallowing saliva that retains the taste of salt after rinsing with saltwater does not invalidate them due to the difficulty of avoiding it. Scholars have explicitly stated that remnants of food carried by saliva do not invalidate the fast. Thus, if food remains between a fasting person's teeth and their saliva carries it, and they are unable to distinguish or expel it, they do not break their fast by swallowing their saliva mixed with it, even if they intentionally swallow it, due to their excuse.