Need to know about registering the birth of my baby?

Just make sure if we can put our daughter surname "Jones"? Because my surname is Ford and my partner's surname is Mason-Jones. We not married yet, will married in the future.

You can give your child any name you want. There's no law saying your child's surname has to be the same as one of their parents or even used by any other person who has ever lived on this planet. There are no legal requirements beyond the same as for first names, ie the same criteria like not offensive, made from letters and not symbols etc.

Yes you can give your child either last name even if you are not married.