It is absolutely normal for an adult child to want to spend time with a parent. You sound absolutely lovely. and very patient with your mum who is, franky, acting like a lovesick teenager but worse. I'd love to give you a hug.
Leaving you home alone at Christmas / New Year was appalling behaviour. The fact you are an adult is irrelevant there. Who would leave anyone they live with home alone for Christmas? Especially their child.
Your expectations are aboslutely reasonable. Your mum's behaviour is unreasonable and distinctly odd.
She seems to be in the thrall of this person, but at the same time ashamed of them. I agree with pp - stop lying for your mum. She can't expect you to do that.
If you have a basically good relationship (or did in the past) I think I'd have a try at have a proper chat and pleading with her better nature. List all the times recently that she has refused to make plans with you. All the times she's cancelled plans for him. The times she's walked away mid conversation to answer a call. Leaving you alone at Christmas. Asking you to lie. Tell her how that makes you feel and tell you'd love for it to be different. That might also be the time to tell her you can't continue to lie for her, and that you are worried for her as there must be something going on that she is ashamed of if she has to lie about it.
Of course you know your mum, and you know how this conversation might go down. I can appreciate you don't want to make things worse between you, but as you aren't happy with how things are right now, I don't know how you'll change it unless you have a real heart to heart.
Aside from that, and in any case, please keep in touch with old friends and make efforts to meet up with them. Be open to meeting new ones - hobby and interest groups etc. Are there any relatives you could arrange to see a bit more of? Build up your support / friendship group gradually, so that it will bother you less if your mum continues to be absent. As someone said above, one day I'm certain she will regret how she's acting. I just hope for both yours sakes she doesn't leave it too long.