For the Animals I see the Rascals as closest in their R&B influences, blend of soul and rock, and the introduction of psychedelic influences later on. Both groups had big hits with both outside material and self-penned songs. Both groups had members in the band as talented and important to their success as the lead singers.
America - Prelude Dino Desi and Billy - Beaky Mick and Tich etc. Damned - Dead Boys You can do this. Just stay away from the A list.
In reverse order, so British Band, who had a few Counterparts US Bands: Echo & The Bunnymen—A Post Punk Version of The Doors and The Velvet Underground.
Dire Straits as the British version of Creedence Clearwater Revival 4 piece guitar based bands (at least in the beginning) with a dominant brother that wrote and sang lead on all of the songs
I like Pavement, but they are just a band influenced by The Fall. It makes no more sense to say that they are the "American Fall" than to say Shakin' Stevens is the Welsh Elvis.
Dr. Feelgood and MC5 In short, both were blues-rock/RnR based bands with cathartic frontmen who were pivotal in shaping the punk/new-wave scenes in their respective countries
The Grass Roots - Love Affair Despite the similarity in sound and approach and the success in their respective countries, neither band was able to score a single chart record across the pond.
Thought I'd try my hand at this: The Beatles = Bob Dylan (The Byrds if it has got to be a band) - most responsible for changes The Rolling Stones = Creedence Clearwater Revival - hard rock riffs and solid rhythm section The Kinks = The Beach Boys - introverted depressed genius and family member(s) doing something distinctly UK/US. The Who = Love - Experimental/life-affirming/often brutal rock The Animals = The Doors - organ heavy macho music The Zombies = Simon & Garfunkel - soothing wonderfully written mood The Yardbirds = Spirit - psych hard rock, not great singing and too many instrumentals The Hollies = The Monkees - Beatles "inspired" Donovan = Jefferson Airplane - heady psych-folk/rock The Idle Race/Move/ELO = Buffalo Springfield/Neil Young/CSNY - a genius (or 2) and his bands The Pretty Things = The Velvet Underground - sold nothing, influenced everything Cream = Blue Cheer - psych hard rock
There isn't quite a British version of Tenacious D, but in terms of blending rock, classical and comedy, Bill Bailey is about as close as you will get.
Happy Mondays and Red Hot Chili Peppers. Both had their roots in raw, post-punk in the early 1980s - both were noticably influenced by Gang of Four, too - and brought bass-heavy funk their respective countries' alternative scenes during the late 1980s and early 1990s.