It does sound like the main issue is self-settling for naps. ds2 had the same issue about the same age that he was getting tired by about 8.30am, so I'd just put him down early but encourage him to go through until 10am - he then seemed to do better at the later nap as well. It was only for a few weeks, then he just seemed to be able to stay awake longer and could extend the morning until 9am-ish.
Can I ask what your naptime/bedtime routine is - do you do anything specific to settle her into her cot or wherever she sleeps? I can't say I'm the expert on this, because we've been having issues with the LTN recently too! Just seem to have cracked it again over the last couple of days.
A couple of ideas to try, anyway: try not to go up immediately they cry on waking early from a nap. Wait 5 minutes and see if she manages to resettle herself - I found I often went up a bit too early and ds2 actually seems to need a short cry to settle himself mostly at the start of naps, but sometimes in the middle too.
Is there something that guarantees her to sleep, even if it's "bad" - car journey, trip in the pushchair, a baby swing? It might be worth a try to get her to sleep the time she needs doing that for a few days until she's less overtired and into the routine of having that long a sleep, then trying to introduce good sleep associations then.
When you're trying to get good sleep associations, consistency does seem to be the key. Doing the same things before you put her down into the cot gives her the message it's sleep time now and helps her to get sleepy. We have a white noise machine as part of our sleep routine, a singsong while having a cuddle/rock, then placed in the cot while I sit nearby for 5 mins, then I leave regardless of whether he's asleep by then or not - only returning if he hasn't settled himself after another 5 minutes. But I don't think it really matters WHAT you do, as long as you do similar things each time as a little mini-sleeptime ritual.
Sorry, this has been a bit lengthy! If I have any other ideas I'll post again later.