How to call multiple mutations at the same time?

I think you can wrap each Relay.Store.commitUpdate in Promise:

    commitMutationPromise = (Mutation, data) =>
      new Promise((resolve, reject) => {
        Relay.Store.commitUpdate(new Mutation(data), {
          onSuccess: (transaction) => {
            resolve(transaction);
          },
          onFailure: (transaction) => {
            reject(transaction);
          },
        });
      }

And commit your mutations as array of promises and catch result with Promise.all(but keep in mind its fail-fast behaviour).
It could be something like this:

      handleDelete = (deleteIds) => {
        const deletePromisesArray = [];
        deleteIds.forEach(id => {
            deletePromisesArray.push(
              this.commitMutationPromise(DeleteMutation, { id })
            );
        });
        Promise.all(deletePromisesArray).then(values => {
              this.onSuccessDelete(result);
            }, error => {
              this.onFailDelete(error);
            });
      }