The Home Office publishes data on the number of asylum applications, initial decisions and appeals, broken down by nationality, in the quarterly Immigration Statistics publication. The latest figures, up until 30 September 2017, are available at the following link: https://www.gov.uk/government/statistics/immigration-statistics-july-to-september-2017
Data for Q4 2017 is due to be published on 22 February 2018.
Table as_01_q (asylum tables, vol. 1) contains quarterly breakdowns of the number of asylum applications and grants at initial decision, for main applicants.
Table as_14_q (asylum tables, vol. 4) contains quarterly breakdowns of the number of asylum appeals allowed. The appeals allowed figures may include a small number of cases where the asylum application was granted at initial decision, but later appealed against. This may be due to the circumstances of the grant (i.e. discretionary leave/UASC leave rather than asylum).