I support the highest standards of animal welfare so I am pleased to note that the importation of fur products is already tightly regulated. As you note it is currently illegal to import furs derived from cats or dogs, or products made from them; in addition the fur and skin of endangered animals or fish cannot be imported without a valid permit.
Meanwhile, under European regulations it is prohibited to import furs or fur products from 13 wild animal species originating in countries where they are caught by leg-hold traps, or trapping methods that do not meet international standards of humane trapping. Strict rules are in also place in the European Union to ensure that animals kept for fur production are kept, trapped and slaughtered humanely.
Once the UK has withdrawn from the EU it will be for the Government to consider future policies, taking into account the outcome of exit negotiations. I am certainly not aware of an intention to lessen the existing standards, and will be interested to monitor the position as it develops.