"I Am John's Pancreas", the debut album by A Witness.
(Start with the obvious bits, eh?)

'Hands' (Little Boots)
Badfinger
Spooky Tooth
'Bootylicious' (Destiny's Child)
Lipps Inc
Chinn (and Chapman) - apologies :-)
Legs (ZZ Top)
'Love In My Veins'(Pretenders)

Gordon Lightfoot
Horslips
Jeff Healy
Joan Armatrading
Kate Bush
Lynard Skynard
Rolling (kidney) Stones
Small Faces

Under My Thumb - The Strolling Bones
Little Willy - The Sweet
Devil's Elbow - Heads, Hands & Feet

misty in roots
toetoe
nine inch nails

tuung n' chic (sorry)

brain salad surgery
bat for lashes

EDIT:

and of course "The F**king C*nts Treat Us Like Pr*cks" by Flux Of Pink Indians.

Little Feat?

Nipple Feat! With a Medicine Head!!

by The Decemberists.

The Necks

No one's mentioned Head, Hands and Feet yet - and they'd be matching.

There was definitely a band called The Toes.

and
Brian Eno's A Year with Swollen Appendices
Perfect Skin - Lloyd Cole and the Commotions

bringing the Faces but he could also bring Heads along too.
Art Garfunkel could supply an Eyes option and Jellyfish could provide a Bellybutton. Rory Gallagher does a nice tattoo too and I'm sure some part of Bono might be useful. And dear old Max Bygraves would, as ever, be advising during the op with 'You Need Hands!' and PP Arnold would be shrieking 'The First Cut is the Deepest'.

These 'creations' are usually dressed in ill-fitting clothes but I think Jack Bruce knows a fine tailor although he would, no doubt, be telling us 'We're Going Wrong'.

a Third Eye Blind and a Third Ear band

They could be "Short People".. or "They Might Be Giants".

Gene Clark
Suzanne Vega/DNA
Membrane

That It Bites song.. Colon All The Heroes..
The Femur of the Species by Space
Anything by Pelvis Costello
In A Womb by Dodgy
Paperback Ureter by the HJHs
Sternum Deliver by Adam And The Ants
Vagus by Sleeper (Neuroanatomical joke..)
Ileum, It Was Really Nothing by The Smiths

From somewhere near Cadiz, there came the sound of a creature stirring..

Red Box?

(....coat)