As a member of the Grey Line Faction I must comment on this.
The magazines will not be long enough, no use arguing otherwise. The horizontal stowage would entail long magazines for stowage and assembly. Don't forget the missiles need to be assembled before they are launched apart from a few ready-use rounds at the front. The Counties had extra length and complex bits not for nuclear resistance but for fitting the missiles with nuclear warheads. This added further weight to the ship. There were schemes to refit some of the latter Colony Class cruisers with Sea Slug aft but it became clear the work would be too expensive and the magazine too small to be worth it (even with two Type 901 radars). Your design would mean gutting the entire forward lower superstructure deck as far as the first funnel, the aft magazine would need to strech beyond the after funnel which would entail some dificulties. This would amount to probably enough work and cost as to build a new-build cruiser.
In our RN AU me and PB felt to get anything like enough capability to engage four targets would need at least a production triple-missile launcher as fitted to Girdle Ness and more than double, perhaps two-thirds, more missiles than the County.
There were studies of vertical magazines for the Sea Slug, that is to say the missiles were still horizontal but were stacked like a multi-storey car park.
Here is my AU version based on a real early study for the County Class. The blockhouse near the Sea Slug launcher hold 24 missiles but is four decks deep! Two above the upper deck and two below. An immense structure for just.
Even assuming your cruiser had 48 missiles in such hangars they would need armouring and would serious affect topweight (as would four 901 radars, I seriously doubt the Belfast could take as much topweight as you have added already). It would be impossible to put them inside the current 6in magazines as they would be too narrow and the structure needs to be above deck to feed the launcher.
Also you've left the belts on the Belfast but they would be useless protecting empty magazines. The deck armour is also useless being below the missle magazines too. Realistically you need to start from scratch from the keel up to get anything remotely workable. At best I feel a two-channel ship might work but 4-channel seems almost impossible unless you want an unaffordable massive vulnerable ship.
[Thanks Thiel, link fixed!]