HOW IT WORKS
There isn't equipment-type Innocence, only parasitic--but that doesn't mean that all Exorcists eat like Allen or are as affected by it as he is.
In Myth!verse, Innocence doesn't just sit in one area of your body, or change only part of your cells. It works itself through your entire body, changing a normal person into an Exorcist and a mythical creature.
However, that doesn't mean to a Noah they would seem 'made of Innocence', the Innocence changes their body and moves on. It doesn't sit in certain spots, even if they've been changed. For example, an Exorcist might have gills, but if a Noah were to touch the gills, it would be just like touching normal skin.
The Innocence tends to settle in certain spots of an Exorcist's body where the most change has happened, or where the shift to weaponize their new bodies occur. An example would be Allen's Innocence has settled in his horn and hooves--this doesn't mean they're separate Innocence, but that he has two places for a single Innocence to develop.
These places where the Innocence settles is more like a canon Exorcist's weapon to a Noah. If they get their hands on one of these areas of the Exorcist's body, they could technically crack/destroy the Innocence.
Amusingly losing Innocence and reverting back to a normal human happens a lot quicker than changing into an Exorcist in the first place. Only reverting hurts so much more.
ACQUIRING INNOCENCE
Some people, like Allen, are born with the Innocence--or like Timothy, have it forced into their body.
However, most acquire Innocence from a traveling general. The Innocence reacts, similarly to in canon when a cube zipped off to find Chaoji in the Ark. However, unlike in canon, aside from a few bursts of power at the beginning, the actual change doesn't occur for a day or so--sometimes waiting up until a week before the change begins.
THE CHANGE
If you're born with the Innocence, generally there are signs of it (ex. Allen's hooves) but the real activation of the Innocence won't occur until about puberty (or slightly before it, as with Allen's trauma). With such Exorcists, it's not strange for the change to be much slower and come in stages--but at the end be much more powerful.
If you acquire Innocence later in life, such as swallowing a piece, or it merging into you from a general, then the change will usually start within a few days up to about a week.
The change is very frightening for many Exorcists, since it seems that they are suddenly becoming something totally different than human. The Order retains that an Exorcist is still human, as their DNA doesn't change, but that doesn't mean they appear, or act human anymore.
It's hard and near impossible to tell how the change will begin, or what type of Innocence an Exorcist has within them. Hevlaska is one of the few who can give a guess, at the very least. Some Exorcists start feeling ill as their insides shift, or itchy as their skin gives way to feathers.
A change into a full-fledge Exorcist can take up to a month--although it's not unheard of to just suddenly wake up as something else within the span of a night. There are also cases where trauma sped up the change--such as an Exorcist nearly drowning suddenly changing into a merman. On average, however, a change usually finishes by week two.
This doesn't mean the Exorcist has any idea how to use their Innocence, however, or even how to do day-to-day things with their new body.
TRAINING AND THE ORDER
The Order supplies for the new Exorcists, getting them what they need to be comfortable while an appropriate general is usually assigned to teach them how to deal with everything that's happened to them.
This can range from archery classes, to learning how to eat for their new body, to just learning how to walk/get around again.
Many Exorcists require odd diets, so Jerry's got his hands full preparing everyone's food. Chairs and beds sometimes need to be designed as well--or reinforced for their strength and/or weight.
Otherwise it's basically how you would imagine canon training while apprenticed to a general.