Hello,
as far as I understand the Geneva Refugee Convention, each country which signed it, must issue documents for you if you have you residence there, especially if you have no possibility to make is elsewhere. See the Text der Convention:
Article 27 - Identity papers
The Contracting States shall issue identity papers to any refugee in their territory who does not possess a valid travel document.
Article 28 - Travel documents
1. The Contracting States shall issue to refugees lawfully staying in their territory travel documents for the purpose of travel outside their territory, unless compelling reasons of national security or public order otherwise require, and the provisions of the Schedule to this Convention shall apply with respect to such documents. The Contracting States may issue such a travel document to any other refugee in their territory; they shall in particular give sympathetic consideration to the issue of such a travel document to refugees in their territory who are unable to obtain a travel document from the country of their lawful residence.
2. Travel documents issued to refugees under previous international agreements by Parties thereto shall be recognized and treated by the Contracting States in the same way as if they had been issued pursuant to this article.
I would anyway recommend that you find a councelling center in the UK in your area, which hat expertise in migration and refugee topics. It is very important to have help and sufficient and reliable information sources in the country and in the place where you live.