In case it helps anyone else, using the new Microsoft Web Service Reference Provider tool, which is for .NET Standard and .NET Core, I had to add the following lines to the binding definition as below:
binding.Security.Mode = BasicHttpSecurityMode.Transport;
binding.Security.Transport = new HttpTransportSecurity{ClientCredentialType = HttpClientCredentialType.Certificate};
This is effectively the same as Micha's answer but in code as there is no config file.
So to incorporate the binding with the instantiation of the web service I did this:
System.ServiceModel.BasicHttpBinding binding = new System.ServiceModel.BasicHttpBinding();
binding.Security.Mode = System.ServiceModel.BasicHttpSecurityMode.Transport;
binding.Security.Transport.ClientCredentialType = System.ServiceModel.HttpClientCredentialType.Certificate;
var client = new WebServiceClient(binding, GetWebServiceEndpointAddress());
Where WebServiceClient is the proper name of your web service as you defined it.