Each country administers its own nationality and whether you are covered by it or not. If both countries permit multiple nationality, then you retain the nationality of both countries.
But be aware of the
Master Nationality Rule; if you are a dual/multiple national and physically in one of those countries, you can be treated as if you were solely a citizen of that country.
Essentially, if you travel back to the country of which you are a citizen and from where you fled for asylum, the UK will not be able to afford you any consular assistance in that country. You could be treated as if you were solely a citizen of that country. The recent famous case of
Nazanin Zaghari-Ratcliffe is a case in point.
This, by the way, is not unique to asylum cases. A dual US-UK citizen could be denied British consular assistance if they are arrested in the US itself, a country of which they are also a national.
Similarly, till the 1970s,
Americans of Polish descent visiting Poland were subject to all the requirements of a Polish citizen, including military service, even if they were not carrying a Polish passport. In the middle of the Cold War, the US and Poland negotiated a treaty exempting such dual Polish-US citizens from certain aspects of Polish law in 1972.
So, although you retain the nationality of the state that you fled from claiming asylum, if you travel there, be aware that the UK government may not be able to provide you assistance. Travel at your own risk.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.