With the old Wilex fuses you have it is easy to strain the fuse wire when tightening it up, so it looks good in centre but has failed at the end.
If following the rules, because the maximum fuse for a ceiling rose is 5 amp, even though you are allowed 16 amp for lighting in general, it is normally 5 amps for lights. You show looking at a 15 amp fuse (blue) not a 5 amp (white) so have you looked at the right fuse.
Today we are not allowed rewireable fuses where it is in control of an ordinary person, this means not so easy to buy fuse wire on the cards like in the past, it tends to be sold on a reel where you can get it so it's a real pain.
That fuse box has had 3 upgrades, first they introduced a cartridge fuse, then a trip with button, and then trip with switch, they are easy to fit, however the circuits should be tested before the up-grade.
So turning off the main isolator because behind the base there are live buzz bars you can fit one of
these [/img]as I have said really should be tested first so I would only renew the faulty one to start with
there is also the 16 amp version however today the fuse box is no longer used, we use a type tested distribution unit called a consumer unit, these now have extra safety features called the RCD, to replace all the fuses with MCB would cost you over £120 which would go a long way towards fitting a new consumer unit. Renewing one at £10.06 where you have the problem may be a way around lack of fuse wire, keep you going until you can get an electrician, but I would not swap the lot as they will be scrap when consumer unit changed.
No children and no DIY and likely you can get away for years without RCD protection, but the day you buy a masonry drill is the day to consider getting a new consumer unit fitted.