I think its Jane. Here is why:
The fact that the musical box is what Lucy last heard so her death suggests she must of died in her room on Good Friday.
Who had very easy access to the Beale house? Jane.
Emma figured out who did it when Patrick had fallen down the stairs in his own home. Lets suppose this is what Emma realised, that Lucy was murdered in her own home.
Emma then examined the Square most likely judging distances between houses.
Emma was still interested in the video of Lauren going to Lucy's house furthering the evidence Lucy was killed at home.
Jane says she saw Lauren going to the Beale house then left without entering.
If we assume this map is accurate, it would of been very difficult for Jane to of even seen Lauren do this and she happened to be looking out the window in a very awkward manner or happened to go outside at this time. My guess this was a cover and she wasn't even in the Masood house at the time and was moving Lucy's body.
The musical box returning to the Beale house was done as Jane had it cleaned and kept it rather than disposing of it for sentimental reasons. I believe Jane didn't mean to kill Lucy. She also seemed to know secretive spots in the Beale house like loose floorboards where she is storing Lucy;s phone. So keeping the box hidden until Christmas isn't past her.
The live episode itself is Ian and Jane's Wedding - if it was any other character it could realistically be any day and Ian finding out at the registry office, saying his vows, is obvious. As something big to either the broom or bride always happens on their wedding day to them personally.
Finally, Emma's phonecall. Emma said 'I'm glad I got a hold of you'. This suggests Emma had doubts they'd even answer the call knowing who it was. Jane was the worst to her that day.
Despite thinking its Jane, I do believe it was an accident - hence why Emma said its still murder to her in the park.
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If there is anything confirmed false in the above, please let me know as I have my facts wrong.
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