Hi there!
Lucy is just adjusting to her new world. Everything that is going on is new to her and her crying is probably her way of asking for reassurance. Lucy doesn't know the difference between day and night yet and you have to help her distinguish. Baby's cry for lots of reasons, hunger, boredom, attention. She is probably sleeping for a total of 16 hours in 24 hours but because its broken up and you are tired it feels like she hardly sleeps - this will settle down.
How are you feeding her? Is she crying because she's hungry? Also, she may have colic and there's no miracle cure for that other than time and patience! Speak to your Health Visitor.
My daughter is now 15 weeks old and I had exactly the same problem with trying to get her to settle into some kind of routine. She was feeding every 1 - 2 hours and hated her moses basket - I slept with her on my chest for the first few weeks!
I am bottle feeding - I made sure she took more feeds by taking my time - if she stopped feeding I would change her nappy, let her rest, wind her and then give her some more - 9 times out of 10 she would take more bottle which meant I could start stretching out her feed times and in turn make her sleep for longer stints. I did this by increasing it by 5 or 10 minutes at a time - I sang to her, walked around the house into different rooms, I even went into the garden with her - anything to distract her and encourage her to take more bottle!
The moses basket is another story! That did take time. After being in my tummy for so long I think the huge space of the moses basket was overwhelming for my Ruby. In my lovely warm arms one minute then thrust into open space then next and expected to settle! So, I kept putting her in it when she was awake and in a good mood and reassured her, then left the room for a few minutes and kept doing this every so often. Eventually she grew more happy to stay in there. Ruby also didn't/doesn't like being fully swaddled, therefore, I half swaddle her - under her arms so her tummy and legs are secure - she seems happy with this. She will now sleep happily through the night in her moses basket after we have the same bedtime routine each night so she knows its big sleep time - kick about without nappy, bath, quiet play, bottle, wind then shush and pat until she is dozing and I put her in the basket, slowly and just shush and reassure her,she eventually goes off but it can take up to an hour sometimes for her to drop off by herself.
I dont leave her to cry because I dont feel it benefits her or me. So I let her cry only long enough to know I wont always pick her up straight away but never too long for her to work herself up into a state. I have found this has worked for us, sometimes when she cries now and I only have to shush and gently rock the basket for her to settle herself again.
I still struggle with her daytime naps - she just doesnt like sleeping that much in the day. She prefers to take her naps in her bouncer chair and to be honest if she's happy to nap in there during the day then so be it! I do try to structure them a little by settling her after her feeds and change ie at her 11am feed and change she usually has a little nap around 12 noon for 30-45 minutes. This has all taken time though and each day it can change dependant on her mood.
It is hard, your not alone with feeling like that, but you must speak to your midwife/health visitor if the crying is getting too much - there has to be a reason for it. I hope some of this may help you. Motherhood is one of the most challenging experiences any woman can go through. We are amazing!