To specialise in immigration as a legal executive, it’ll help if you can find a job with a firm that deals with immigration and asylum issues, whilst training. You can find more information about qualifying from the Chartered Institute of Legal Executives.
To specialise in immigration as a barrister or solicitor, it would be useful if you could do some of your training in this area of work. You can find more information about solicitor and barrister training from The Law Society and The Bar Council.