There is no reason why you shouldn't be able to progress on low carb and a caloric deficit if done right.
We can't say that you are eating not enough if we don't know how much you are eating. Try not to go lower than 20% of your daily needs or your body will start to punish you. Not being hungry is a good indicator that you can't be doing it that terribly. Are you tired all the time? That could mean amongst other things that you don't eat enough (of the right stuff). Getting weaker? Another possible indicator. The only way to be sure is to count calories. That has gotten very painless thanks to smartphone apps, consider start doing it if you haven't already.
I'm not the biggest fan of low carb because:
- Carbs are great, especially for sport
- Low carb is made for short-term weight loss. You still have 40 years of life ahead of you, can you not eat carbs for 40 years? That sounds unnecessary hard for almost 0 benefits.
- You have to sacrifice great (sometimes called healthy, but I hate that word) food but still are allowed to eat garbage if it's low carb instead of learning how to eat a balanced diet. Anything that tells you to not eat lentils is doing something wrong if you ask me. Also, a lot of carb stuff tastes great!
But I wouldn't agree with those that say you absolutely have to eat carbs. You have been doing great so far regarding your weight loss goals, why not continue? I'm just saying if you have a good reason to do it, do it. If you're just doing it because it's the latest fad, ask yourself if another option might be better suited and leave you with more energy. You have to keep in mind what happens once you have reached your target weight.
One cannot also say if you work hard enough (or maybe too hard) because that information is also missing from your post. Do you do 6 pushups and then collapse on the floor and can do no more 3 times a week or more?
You're not that heavy, but still you have to start lifting a lot of weight with exercises designed for guys ~25 pounds lighter and 15 years younger than you. Also, remember, as a kid, you would be doing all kinds of athletic stuff all day. Those muscles might have been inactive for a decade. The same is true for your coordination and your nervous system might also have forgotten a couple of things. The great thing is that you started using all of that again.
The problem with bodyweight training is that there is of not a clear progression as there would be with weights.
However, you can still do a progression, you just have to use your head a bit: Instead of pushups, do inclined pushups for example. The larger the angle, the easier it gets. Try to get ~ 3x12 done maybe 3 times a week. If you have to almost stand up, do it. Decrease the angle until you get better at it and at one point you'll be doing standard push ups or better. You will find such progressions for basically any exercise.
Btw, I assume that you have made a good training plan and train your entire body, avoiding imbalances and so on. Oh, and btw you are > 30 = you might have a job and so on. If being cheap is the reason why you are doing bodyweight stuff (as I said, as an heavier and out of shape person, dumbbells and so on might be a better idea. You can more easily progress) - don't be cheap. But if you have better reason, stick with it.