Some rabbis prohibit the husband’s attendance in the delivery room altogether. Among those who permit it, we have not seen this issue discussed directly. Based on their reasoning, however, cutting the umbilical cord would not present a halachic problem so long as you are covered sufficiently (e.g., with a sheet) for the minute or so it would take to cut the cord. This consideration should be discussed in advance with your medical team.
As for whether cutting the cord would be considered touching, we question whether a placenta on the verge of being detached is really considered part of your body. In any event, your husband would not touch it directly, but with a scalpel.
For further details on the halachot involved in the husband’s attendance in the delivery room, see the relevant section in our article on Childbirth.