Hi and welcome to the forum John. When I read your post I saw myself one year ago in the exact same period. I started in September at 240 lbs and 5’7 height. The numbers were almost like yours:
0 pull-ups (as a matter of fact I couldn’t dead hang for more than 5 seconds)
Till now I’ve lost 55lbs and 15cm from the waist and I’m planning to lose another 20 lbs. I didn’t kept any diet (except the fact that I eat clean and unprocessed food) because I didn’t wanted to lose the weight very fast and remain with the loose skin. Even without dieting I lost the recommended 1lb/week, which is considered the healthiest way to lose weight (loosing the weight too fast might cause you other problems).
So, long story short, you’ll have to stick with the basics and increase the numbers in pu’s, pulls and dips. Also don’t neglect your core and legs (especially legs), but be careful with your knees (I’ve destroyed mine because of the over-usage along with the overweight). After that, you can start with the more enjoyable part of calisthenics like levers, flags, MU’s etc. You can also do cardio if you like it. I’ve never did, because I stick to the idea that the more muscle you have the more calories you burn, plus the fact that I’ve always hated cardio. I kept the numbers low (up to 5 sets of 5 reps) and if it became too easy I’d change the leverage or the exercise.
There is no shortcut or “secret ingredient” to losing weight, just plain hard work and dedication. To make it more enjoyable you’ll have to view it as a lifestyle. At the beginning I had to motivate myself to start a workout, but now it comes naturally and I even miss it when I’m not able to do it.
Sorry for the long post (I guess I got overexcited when I saw someone just like I was and willing to do what I did) and for any eventual grammar mistakes (I wrote it in a hurry).
Also, if you need help with progressions don’t hesitate to ask.