It can be done
I started off at the first of the year in the same boat. I was 50 yrs. old, fit for running and cycling, but couldn't swim 10 meters without stopping and being completely out of breath.
I focused on swimming during 13 weeks of lessons and practice, making sure I got myself to the pool at least 4 times per week. The first thing was to learn balance in the water. Everything else came easier after that. Breathing was still the hardest thing to learn. Ironically, breathing came easier after I tried to learn on the "other" side.
I was able to swim 1200 meters in just under 30 minutes after this 13 weeks. I'm gearing up for a half IM at the end of the summer.