In the interests of fairness I should first say I'm Nc with my sister.

"My younger sister dumped me by email about 10 years ago after an argument." yet later you say there was 'no reason' here's at least one
"We had a difficult upbringing with an alcoholic mother who often neglected us and put us at risk. My older sister was of an age where she could go out and escape so it fell to me to take care of her quite often. As a result she felt more like my child than my sister." that's down to your mother not her

"The pain at the time was unbearable. I actually did contemplate taking my own life quite seriously." I doubt she was finding it easy either
"When she fell out with me she was pregnant and I thought she might just be feeling emotional/finding it difficult and things would sort themselves out" well that's quite dismissive of her feelings, just because she was pregnant at the time doesn't mean her thoughts and feelings aren't valid
"but she would not talk to me properly about what the problem was and still hasn't/won't." again claiming you don't know yet in the next sentence "As far as I know, she fell out with me because I was cross with her about something she did that was pretty thoughtless. Just normal stuff really." in your opinion, clearly she disagrees
"So it seems maybe it was an excuse to cut me off and this had been building for a while." or it was the final straw
"What's really confusing though is that she had always been really affectionate and loving, coming to my house regularly, telling me all her problems, sharing her successes etc." no relationship is all bad, but when the bad outweighs the good it can be better to cut your losses

"There was a fashion around then to go 'no contact' or call people toxic and cut them off and I wonder if this was part of that. She was going through some stuff herself and I took the brunt?" again dismissive, not only of your sister but of others like myself who have made the very difficult decision to go nc

"She said at the time I was a bully. That really hurt. I definitely had too much of the parent in our relationship but bully's want to hurt people and I would have taken a bullet for her." bullies rarely recognise it in themselves. Bullies generally don't have a desire to hurt, but to control. Bullying behaviour generally is borne of fear and anxiety. Fear of what will happen if things aren't done their way. Anxiety of losing a relationship
"If it I was true that I was bossing her around, which is entirely possible why was she always visiting me and asking for advice?" so you even tentatively admit you were 'bossy' yet are still dismissive of her feelings on this. Why still ask your advice? Because consciously or subconsciously you've trained her to
"I'd love to have an actual proper talk about this maybe with relate? But she doesn't was to resolve the problem and doesn't want me it seems." that is her choice

"The first year or so I tried to keep some contact going with cards and presents etc." had she asked you not to?
"but after bursting into tears in the post office while wrapping a Christmas present for my little niece who I'd only seen once or twice I wondered if I ought to start protecting myself from the pain. Since then I haven't bothered and always hoped she might contact me at some point but ten years have passed." her pain matters too

"She once sent me a business card to tell me that she's set herself up as an 'energy healer' I think because she thought I might need some healing (?) this made me feel slightly better because if she's a bit of a crank maybe I don't need to take it to heart so much but still, I miss her." i don't personally believe in energy healing, but her beliefs are no excuse to dismiss her feelings
"Or I miss who I thought she was, because maybe I was wrong about that?" or maybe she just isn't who you WANT her to be
"No idea whether to just let the situation continue or try to resolve it. More pain for me maybe? She doesn't want to see me or she'd ask." there's your answer, assuming she could contact you if she wanted to


"Why not just say "stop micro managing, stop telling me what to do, I'm not a kid, it really does my head in' the. I would have had a chance to correct that behavior, which is after all just a left over from my having to take responsibility for her when we were kids." are you absolutely certain she didn't try to and you didn't hear her? That's fairly common in dysfunctional relationships


"Still and Huldra, I tried asking my other sister to intervene, explaining how distressing I was finding it but it's not a supportive family (apart from me and my younger sister, or so I thought)" or your older sister agrees with her. Are they still in contact? I'm guessing yes


"Yes, I did apologize at first, I should have explained that. I managed to get her on the phone by persisting, which she didn't like," that could be viewed as harassment especially if she specifically asked you not to contact her
"and asked her if she could explain exactly what it was that I had done wrong. She brought up a different argument from the past, which is a long story, when she was about to spend her life savings on bailing our mother out. I didn't think this was a good idea as I knew our mother would just end up in the same situation down the line so it would be her hard earned money wasted. It was an extremely stressful situation, long story. DS didn't want to listen to my advice (that she'd asked for) and I lost my rag and shouted at her down the phone." her money her choice, again shows that she didn't go Nc over one argument, one incident


"In my apology I said, yes you are right, I shouldn't shout at you, I'm sorry for that. I couldn't have been any clearer. Didn't make any difference at all. In fact it only confirmed her belief that I am evil and she is lovely it seems to me, from what I heard afterwards." because it wasn't just the shouting? Because you don't have the right to tell her what to do. Whether or not that involves shouting. You could have advised, or said you didn't think it was a good idea, but you cannot tell her what to do
"Just wanted to add, I do think there was a fashion a while ago, perpetuated by internet chat and online cod psychology forums around cutting people off and then being very smug about it. These things are part of culture and emerge in movements. I have fallen out with people badly myself when I was younger and felt very self righteous about it so I know where this is coming from." Please STOP with this nonsense it's incredibly offensive. I don't know anyone who's gone Nc either in real life or online that has made the decision lightly or without great discussion and thought and usually the support of others in real life including therapists


"I have had quite a lot of professional psychotherapy and also was a member of a self help group for 7 years so I'm not without self awareness." how do you know your sister hasn't also done similar?
she could well have had several years of therapy too
"I loved my sister so much but I'm not sure I love her now. This is why I am so conflicted. I'd like to resolve the problem and have some limited contact just for practical reasons" what are the practical reasons? If she's still in touch with your older sister surely your older sister can deal with her on any family issues
"but I can't think of anything crueller than cutting someone off who you know loves you for so many years without any warning or explanation and without any chance to try to resolve the problem. I'm not sure she really is the lovely person I thought she was." there are many things crueller than going Nc, also she DID give you an explanation and warning, that's what the email and discussions about your behaviour were. Also, she may be a lovely person, just not the person you WANT her to be
If she is able to contact you, if she is still in contact with your older sister then my advice would be let your older sister know that you are open to hearing from her, that you are truly sorry for your past actions (only say this if you are please) and would welcome the chance to resolve matters.
Other than that leave her alone.