If UK and Ireland are both members of the EU, why does UK not have a scheme for researchers from third countries (http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=URISERV:i23023), while Ireland does (http://www.inis.gov.ie/en/INIS/Pages/Researchers)? How can third-country researchers in the UK encourage the UK government to introduce such a scheme?
The UK (and Denmark) did not opt-in to this directive.
The UK, Ireland and Denmark have the right, under Treaty, to opt out of certain directives. Ireland chose to opt-in to this directive. The UK and Denmark chose not to. You can lobby your government to put diplomatic pressure on the UK. I would not expect any foreseeable government to change this.
There exist alternative schemes for third party researchers: Academic Visas and Tier 5 admission, for which see the travel stack exchange.