score:6
As traveller has posted in their comment, the process would be the same as those who have applied without a last name. The UK visa department has dealt with this issue many times, and will have a process to follow when they see this. In the place of first name, simply put a '-'.
A similar issue was discussed here and commentators have had success with putting '-' in the box for last name, similarly people have put N/A or other statements to indicate there is no last name for the applicant have also been successful. Please see below for one such example:
I had a similar issue with my wife’s passport. I had put '-' in the surname field.
Visa was granted by the high commission in Chennai. In the receipt her name was 'firstname' 'firstname'. However on the visa her name was only 'firstname'
She recently came to the UK, and there were no issues in immigration etc.