Q:

Who established the original F1 desktop BIOS key and why did laptops use a different key?

It seems early on IBM, MS, and DOS clones established a standard of holding the F1 key down during boot to access the BIOS setup. Yes there were a few much less common combinations that used the DEL (perhaps the del key is the most common now) or CTRL+ALT+ESC. 
Who established the early F1 desktop BIOS key and why did laptops start using an entirely different key - usually the F2 key? Who set the laptop standard for the BIOS key? 
If there is interesting history that lead to the F1 or F2 key it would be of interest too.

A:

Quick Answer:

Who established the early F1 desktop BIOS key and why did laptops start using an entirely different key - usually the F2 key? Who set the laptop standard for the BIOS key?

F2 came first, before any use of F1, as F1 was reserved for error recovery. And it was IBM that introduced F2 in the first place. F1 was later on chosen by IBM to simplify handling as both (error as well) ultimately did lead toward BIOS configuration. Except, at that point many compatible manufacturers had already chosen to follow IBM's original decision with F2.

Yes there were a few much less common combinations that used the DEL (perhaps the del key is the most common now) or CTRL+ALT+ESC.

This might again depend on your personal exposure to certain brands and computers, as the early majority was using multi key combinations, which soon changed to DEL. F1/F2 are rather confined to brand machines.

History

If there is interesting history that lead to the F1 or F2 key it would be of interest too.

Here you go:
First of, there is no standard, at least none set by IBM, as their way to access BIOS setup was thru a separate setup disk. It was when BIOS manufacturers enhanced their BIOS to include setup, each picked whatever they liked. And more important, each board manufacturer/system builder could/did modify this according to their ideas. When buying a BIOS customizing the setup key (and others) is one of the first options that where offered.
Originally  the PC's setup was done via DIP-switches. When the AT introduced a battery backed up CMOS RAM, setup became a software, supplied by IBM on a floppy named ''Diagnostics for the IBM Personal Computer AT'.
Phoenix, who presented the very first clean room BIOS in 1985 (*1), offered the first AT compatible BIOS in 1985, still with a similar disk. Soon the BCU (BIOS Configuration Utility) got integrated into the BIOS ROMs. The first keystroke used to reach setup during POST (Power On Self Test) was CTRL-ALT-S. The key combination was chosen to minimize conflict with other utilities, and configurable by the licensee (*2).
Award followed with CTRL-ALT-ESC and AMI later on with DEL. Which then got taken over by most manufacturers as AMI was quite successful with board manufacturers not at least due a low price. Phoenix got their main market with brand systems, likewise Award (*3)
When IBM introduced the PS/2 systems they finally also included setup into their BIOS (*4). Here F1 was still meant for error recovery if some part of the BIOS couldn't act as configured. That included not only the infamous "Press F1 to continue" on a non existent Keyboard but every error during POST (*5).
With build-in setup, most errors resulted in a switch to setup, effectively F1 and F2 went at some point into the setup screen, which let IBM join them for simplified handling (and manuals) - right when many compatible manufacturers have switched from DEL (or whatever) for F2. So in the end even IBM/Lenovo had to go for F2 again - isn't it ironic?

Common uses are/have been:
(Yes, I have a quite torn up sheet with all the variation I encountered over the years. Written in pencil and touched many many times, so it's quite hard to read in some places:) *6)
F1

BIOS default:

Older AMI
MR.BIOS

Older Acer
Some Dell
IBM/Lenovo
Older Toshiba
Packard Bell
(older) Gateway

Fn-F1

Some Dell Laptop
Lenovo Laptops
Some HP

F2

Mainboards:

Intel mainboards
Asrock mainboards

BIOS default:

Older AMI
Inside BIOS
DTK BIOS
ALR PCI-BIOS (80486)

Many Asus brand computers
Acer
Fujitsu
SIEMENS
Most Dell
Some HP
IBM
Lenovo
Sony
Sharp
Samsung
Most Toshiba
NEC
Gateway

F3

Some Dell
Sony

CTRL+ALT+F3

Lenovo (pre-IBM)

F9

Asus

F10

Compaq
Some ASUS
Standard on most HP
Samsung

CTRL-ALT-F11

IBM Thinkpad - press when in DOS (!!!)

F12

Some Dell
Toshiba Equium series (joint with boot menu)

CTRL-ALT-S

BIOS default:

Early Phoenix

DEL

Mainboards:

ASUS mainboards
ASUS brand laptops
BFG mainboards
Biostar mainboards
Foxcon mainboards
Gigabyte mainboards
MSI mainboards
EVGA mainboards
ABit mainboards
Shuttle mainboards/barebones 
Soyo mainboards
Super Micro mainboards/serverboards
Tyan mainboards

BIOS default:

AMI
Award
Phoenix

Packard Bell
Acer

CTRL-ALT-SHIFT-DEL

(can't read anymore)

INS

Some ASUS

CTRL-ALT-INS

BIOS default:

Some Phoenix

Lenovo

TAB

(Can't read)

ESC

Many early Toshiba
Many HP

Fn-ESC

Some Dell Laptop

CTRL-ALT-ESC

Older ASUS
Older Acer
Shuttle barebones
BIOS default:

Early Award
Some Phoenix

CTRL-SHIFT-ESC

Tandon PC

CTRL-ALT-ENTER

Some old Dell

Special Keys

Sony VAIO

Assist press when powered off

Microsoft Surface

Volume-Up
Volume-Down 

Lenovo/IBM

NOVO button
ThinkVantage button (Thinkpad)
or IBM Access button (Pre Lenovo Thinkpad)

As shown, many manufacturers/brands changed their setting between machines or over time. Often when buying boards from a different supplier, or incorporating a new BIOS. HP is eventually the greatest example, using almost every key, including F1, F2, F3, F6, F10, F11, F12 and ESC over the years.
But the general pictures shows well that basic BIOS and thus bare boards are usually delivered with DEL for activation, while most system builders change it for F2 when integrating their logos and driver modifications.

In addition:
F10

Start with default BIOS on ACER
Network boot on many other

F11

Boot menu on Sony and Intel

Well, the usage of other keys and the function they access might fill a book on it's own.

*1 - Well, Compaq's BIOS was before and also clean room, but it was not only a PC one, but also not available to other PC manufacturers.
*2 - Licensing fee for Phoenix BIOS was a whopping 290,000 USD. This not only included configuration support to adapt it to specific hardware, but also an insurance against being sued by IBM.
*3 - Phoenix later on acquired Award - after buying some other manufacturers - so by now Phoenix and Award is about the same, and both based on a a third implementation :))
*4 - Which in fact was written by Phoenix for IBM.
*5 - Who remembers all the beep-codes anyway?
*6 - What's also hard to read is the often displayed "Press <whatsoever> for Setup" Surely there is a short display interval is adapted to the fast reading skill of today's nerds. Older ones like me will have to guess. But there's a secret trick for all of age: pressing SPACE during POST often stops the screen when all checks and setup is done, right before booting so even slow readers get a chance.

