I ended up creating a fixed version of NestedContentMapper to handle this issue. The code is as follows for anyone that runs into this problem.

public class NestedContentMapperFixed : NestedContentMapper {
    private readonly string docTypeAliasValue = "ncContentTypeAlias";
    private readonly Lazy<SyncValueMapperCollection> mapperCollection;

    public NestedContentMapperFixed(IEntityService entityService, IContentTypeService contentTypeService, IDataTypeService dataTypeService, Lazy<SyncValueMapperCollection> mapperCollection) : base(entityService, contentTypeService, dataTypeService) {
        this.mapperCollection = mapperCollection;
    }

    public override string Name => "Nested Content Mapper Fixed";

    public override string GetImportValue(string value, string editorAlias) {
        if (string.IsNullOrWhiteSpace(value) || !value.DetectIsJson()) return value.ToString();

        var nestedJson = JsonConvert.DeserializeObject<JArray>(value);
        if (nestedJson == null || !nestedJson.Any()) return value.ToString();

        foreach (var item in nestedJson.Cast<JObject>()) {
            var docType = GetDocType(item, this.docTypeAliasValue);
            if (docType == null) continue;

            GetImportProperties(item, docType);
        }

        return JsonConvert.SerializeObject(nestedJson);
    }

    protected JObject GetImportProperties(JObject item, IContentType docType) {
        foreach (var property in docType.CompositionPropertyTypes) {
            if (item.ContainsKey(property.Alias)) {
                var value = item[property.Alias];
                if (value != null) {
                    var mappedVal = mapperCollection.Value.GetImportValue(value.ToString(), property.PropertyEditorAlias);
                    item[property.Alias] = mappedVal != null ? JToken.FromObject(mappedVal) : null; // .GetJsonTokenValue();
                }
            }
        }

        return item;
    }
}

I just extended the NestedContentMapper to add in the missing GetImportValue method. Then I needed to remove the existing NestedContentMapper with

composition.WithCollectionBuilder<SyncValueMapperCollectionBuilder>().Remove<NestedContentMapper>();

in a composer (my code is for umbraco 8, so the syntax will be a little different when working in umbraco 9).