how bad this really is will be relative to the size of fish you introduced / overall tank size. 15 small fish in a 125g is likely a non-issue but in something like a 20-30g this is a steep increase of bioload. i will assume your situation is closer to the latter given your readings...
keep doing very frequent large water changes to avoid toxic levels. you really have no choice on this matter. your tank will cycle eventually but you will need to keep a close eye on parameters until stable so your fish stay healthy. keep in mind that nitrites / ammonia are BAD for fish so you really want to minimize them. IMO you should consider doing more frequent water changes (daily). If nitrites are building up over 24 hours you want to get them out of there.
I believe that doing very frequent water changes does not 'prevent' tanks from cycling (I think this is what you are assuming?). people may argue me on that but I don't buy it... i am not sure why so many people think this is the case and think they have to have a constant high level of ammonia for any bacteria to colonize. when i set up a new tank I almost immediately add a handful of fish (5-10% of overall end-game bioload) and perform ~daily 50% water changes for a while then check tank readings every once in a while. I slow down the frequency of water changes over a period of a week or two and then after a little while my nitrites are gone forever so yes, my tank is cycled. then over time add fish in batches of 10-25% of end-game bioload until fully stocked. you never want to make drastic increases IMO so don't do stuff like add 10 fish to a tank with only 10 fish in it.
basically in the future do not attempt crap like this. this is how fish get sick and diseases come in and wipe out your tank (i have been there before). one of those situations happening will convince you it is not worth it in the future. add a few fish and stock your tank over time as the tank stabilizes. you are much less likely to run into issues stocking your tank over time. now i accept the fact that it will take 3+ months to fully stock my tanks given initial cycling time and purchasing/quarantining fish in smaller batches. the survival rate / lack of health issues is by far and away superior when you take it slow.
The vast majority of bacteria is on surfaces, not free floating. You and your fish are better served by the water changes clearing wastes than by whatever little bacteria is floating in the water.
Keep monitoring, and when you get 0s for ammonia and nitrite, and start seeing nitrates rise, you can stop the daily WC.
yes, basically this
That's what I figured but wanted to check. Thanks. The nitrifying bacteria bottle says not to do water changes for two weeks. Would this be because when first added it needs somewhere to attach?
No, you need to perform WC. the 'cycle your tank with a bottle' products are all fine and good but IMO you need to do daily WC to prevent toxic levels in your situation. IMO using these products and not doing WC on a brand new tank for 2 weeks is terrible advice. the bacteria needs to colonize in your filter media / surfaces within your tank and this will not happen to extent you need it to immediately after dumping in a bottle of products. performing WC and having your bacteria build up over time is more reasonable
I use Seachem Stability when I start my tanks (not sure if it actually helps or not but whatever) and do WC like I mentioned above. IMO the method I outlined above is by far the best way to go. your tank WILL cycle eventually and your fish will be happy thoughout. just be patient