Turn then power off. Then check the left hand pair of fuses first. They're only 5A which is consistent with a lighting circuit. (One's probably for upstairs and one for downstairs).
I still think it will be one of the 5A fuses on the left. The fact that the one on the right is sticking out slightly is probably just because it's a poor fit. (The right-hand pair should be for your upstairs and downstairs ring main sockets).
BTW: My house is far from 'ancient'. (1970's Barratt home). but it's still got the same type of fuse board which you've got.
To pull a fuse out you often need to rock it up and down a bit. (i.e. pull alternately at the top and bottom). If it's sticking though, I do recommend turning off the power. If your finger slips behind the fuse as you're rocking it out, it could come into contact with the metal prong of the fuseholder while it's still connected to the mains power.
My house is quite old - about 120 years - and the wiring was dire when we bought it. We were doing the washing one day, and had the immersion on. I turned on the tv and blew the lot. Turns out the bodger who previously owned it ran one line through the kitchen/bathroom and they were all on it. I've since replaced the fuse box with trip switches, find them much safer. The old box didn't even have fuse wire fuses - they were the screw-in bakelite type.