I used to teach swimming lessons, have coached competitive youth swimming, and was darn good at both (much better than I ever was as a competitive swimmer... I was mediocre at best
Here's the thing. Swimming is a skill that kids catch onto at vastly different rates. Some kids love it, are comfortable in the water from the get-go, float independently as soon as someone shows them how, and can do basic propulsion with just a few well-crafted lessons. Others... don't. It's one of those things where a good teacher can guide the process along, can be creative in getting kids to try whatever it is they're afraid of (ears in the water, floating without asssistance, etc.), and can help them out, but some kids, for various reasons (temperament, experiences or lack thereof, etc), don't take to it as quickly. They can most certainly learn to swim, no problem... but it can take time. So often it's not an issue of the kid not being physically capable, or the instructor not doing the right thing... it's that the kid's affective filter (to use some obnoxious educational jargon) is still so high when it comes to whatever aspect is being learned. The best "cure" for this is a mix of (a) time, (b) experience (more lessons if they're fun or just playtime) and (c) a lot of watching other people swim happily. I had one parent of a very anxious child who would come in an hour early and watch the advanced levels (where the kids had zero anxiety and were practicing joyfully), all the time commenting, "look, that girl in the green suit just put her face in the water!" "Oh, wow, that boy over there just jumped in and swam to the side ALL BY HIMSELF!" No comments about "you should do that," just casual conversation.
So yes. Those videos you see of six month-olds swimming independently are real and TOTALLY possible. I've taught those kids. I've also taught kids who, after a year of lessons, FINALLY put their noses in the water (voluntarily, I never forced kids... even the ones whose parents told me to). A month is a short amount of time, but starting lessons, or starting to teach them yourself (PM me if you want specific ideas, I love to geek out on swimming
) is never a bad thing. At that age and at the beginner levels, lessons are more an introduction to very basic skills (ears in the water, blowing bubbles, kicking, paddling, faces in the water, ears in the water on the back, etc...)... and to the fact that being in the water is FUN. They just set the stage for later learning, and throw in a few "survival" skills... but (see below and above) this is no guarantee and varies greatly from kid-to-kid when it comes to how quickly they are able to do stuff independently.
An added random note... Anyone who says they can "water-proof" your kid, IMO, is selling snake oil. Dangerous snake oil, because there have been cases where parents thought their kid was OK because, "They took lessons and can swim!" Small children are never, ever waterproofed. Ever. Period. I always emphasized this at the beginning of my infant lessons... "I can't make your child water-proof. If I say that I'd be opening myself up to a huge liability, and even worse, something bad could happen and I would never forgive myself."
I'm long-winded when it comes to swimming. Like I said, I like to geek out on aquatics.