No reason why it shouldn't be. Westminster Cathedral (note - this is the Catholic cathedral, not the same as Westminster Abbey which you may be thinking of) is a working parish church as well as a cathedral (I've been to mass there one sunday when I was in the vicinity) and so does hold funerals. From their website:
Persons wishing to arrange a funeral at the Cathedral or Sacred Heart Church, Horseferry Road, should ask to speak to a priest at Cathedral Clergy House in the first instance.
I used to go to the Cathedral a lot as a student, and know some of the clergy there well. Pdf is quite correct, any member of the congregation can request a wedding, baptism, or funeral in the Cathedral and it is usually accommodated. You can even arrange to have the (adult) members of the choir sing at a service as long as you pay the fees (they are all working professional musicians). The fact that the funeral was officiated by a bishop and sung by the full choir suggests it was a service that was arranged by the Cathedral Administrator rather than solely at the request of the family.
On the matter of Catholic servicemen, all that would be required would be a request by the family and if appropriate the arrangements would be made.