You either have old-fashioned rewirable fuses, or you have miniature
circuit breakers - MCB's. There's no such thing (or shouldn't be) as a
breaker with fuse wire inside. (Though there are fuseholders with
cartridge fuses inside).

Either way, a rewirable fuse should not be rewired with higher amperage
wire than it is rated for. A MCB can only be replaced. On top of that
the holder a rewirable fuse plugs into is sized for a particular rating
of rewirable fuse - or for the same rating of replacement MCB.

The rated fuse (or trip) value should only be increased if the rating
of the fixed wiring is appropriate - I'd be concerned somone asking
questions about the "fuse wire inside the breaker" isn't someone that
should really be messing with this.