You can also say 'partner'. Like if you said: My partner and I are going to the cinema, it would be taken that you are romantically involved with them.

'Sweetheart' is a nice old-fashioned word that covers both. It's not used much these days, but it does answer your question. 'Partner' works, but it's mainly for adult, more 'settled' couples. It's not something teenagers would use, for example. Some people use the expression 'my/his/her other half' to refer to a partner, but this is also for adult couples. Or how about 'squeeze' ?? I've seen that on the celebrity pages ....

There is not a common gender neutral word for girlfriend/boyfriend. "Lover" has a sexual idea to it, so I would stay away from that. "Spouse" is good if you are talking about a married couple. "Significant Other" is sometimes used, however this is most commonly used for gay/lesbian couples.