Haven't been to Belfast, but have been to London. It's big, a lot of people. I have been in London alone and nothing has happened to me. It was a pleasant experience overall. A lot of tourists and people are generally friendly. Since you speak the language I really don't think you would get lost there. You will definitely not get raped if you stay in the touristic areas. Just take the usual precautions - avoid dark empty streets and don't hang out with men you don't know. Pickpocketing is much more likely than anything else so hold on to your purse

Your mother is overreacting. A lot of young people leave their countries to go abroad to work and/or study. It is absolutely normal. I don't really see the link between traveling and getting raped, you should just get this word out of your head. Furthermore, you are going to civilized places with Starbucks with free wifi on every corner.

I don't know if you already have a place to stay. If not and you are on the budget I would recommend YHA hostels - it's a big chain of hostels. The prices are low, but you have to share the room with other people. They also have nice lounges with wifi and a nice atmosphere.