Gwyneth has not shown any signs that she is an unfit mother during the time that herself and Chris were together. It is rare that in cases where the mother is deemed fit, that a court will side with the father, thus making it unlikely that Chris would be given custody of his children.
Chris' profession as a touring musician would also not fair favourably for him, with long days travelling and working in venues unsuitable for children, it is unlikely that he could confirm that he would be able to provide personal adequate care for the children. Gwyneth however, as an actress who has no future film commitments, could easily show that she would be free to support the children's needs.
Most mothers get custody of their kids when their is a divorce. Unless, the mother is seriously unfit to raise the kids. Gwyneth is a highly paid actress and seems to have her head on straight. Sorry Chris, It looks like you are getting the weekends. That is the way it usually works, unless Gwyneth gives Chris the custody.
The children are currently based in Los Angeles with their mother. Chris has already departed for England. It would be highly unlikely that Chris would uproot the kids and take them to a different continent. Also, the children are young and mother's generally retain primary custody in such cases. It's likely that Chris will visit the kids, but they will not live with him full time. With Chris being a musician, it would also be hard for him to parent full time when he's on the road.
Since this involves a US and British citizen, I don't really know what they'll do.Without knowing specific details of the Paltrow/Martin split, I'd guess that Apple and Moses will stay with their mother. Even if there's a joint custody type of agreement, most of the time, the children will stay with their mother and have specific periods to visit their father.