Without her agreement to take him, you will need a court order.
The most straightforward route would be a child arrangements order for the child to 'live with' you. You would need to apply for permission to make the application first (see section 5 of the application form I'll link to below) as you do not currently have any right to make an application to the court (the only people who have automatic rights to apply are those with parental responsibility, or other orders in place).
You have to try mediation first, to see if you can come to an agreement, but to be fair mediation will not be enough in your circumstances as you will need parental responsibility in order to make any significant decisions about the child's care in the future - a child arrangements lives with order comes with parental responsibility.
The form you need is a C100, which you can find here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/734888/C100_eng_0818.pdf
The cost of the application is £215, unless you are on a low income or benefits, in which case you may be entitled to fee remission, see here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/634329/ex160-eng.pdf
CAFCASS (social workers) will likely be ordered by the court to conduct a welfare report - and CAFCASS may pass this on to the local authority that have previously been involved with your sister to do this report as they have had recent contact (baby being only a year old). The report will make recommendations to the court as to whether or not the child should live with you or whether an alternative plan should be considered.
It would be best to get your sister's agreement before applying to court if at all possible.
Does that help?