This is a hardware limitation of your keyboard. Unfortunately, the only solution is that you must purchase a better keyboard that does not have this limitation.
To be more specific:
In order to save money, keyboard manufacturers often put many keys on the same "circuit" of sorts within the wiring of the keyboard. This prevents multiple keys in the same region of the keyboard from being pressed simultaneously. Sometimes it even prevents more than 2 keys at all from across the whole keyboard being pressed at once. Often the shift, ctrl, and alt keys are not within this limitation, so you can hold shift and press 2 other keys at once and it will still work fine.
Even high-end gaming keyboards often have a similar hardware limitation, although the cap is much higher so that it is unlikely to be reached during the normal course of gaming.
I've been able to work out that Button 0 and Button 1 are actually Left and Right Mouse (may be the other way around)
These can be assigned at the Controls > Customize Controls screen by clicking the Attack Hand/Open hand options and double left/right clicking (as you need to left click to start the assigning process)
Then, by using space for legs and E for head, the main controls sort of work ok.
I'm Currently using:
1-4 Weapon Change (sword, hidden blade, etc)
Space Legs
E Head
Left Click Attack Hand
Right Click Open hand.
That way, I can loot/attack mostly with the mouse. Pressing and holding W (up), E (head) and Space (legs), and directing myself with the mouse, gets me up and over rooftops ok.
I've only gotten to an early part of the game, so I haven't had time to perfect my keys. I'll update this post when I do, and let you know what is the best key arrangement (in my opinion).
UPDATE: The above works well, with Q/F in use for locking, etc.
This should be valid for AC:B too.
Best Answer
Figured it out, You have to press Shift and then a mini menu pops up that you can press 2 or 4 on to quick switch.
Gears 5 conditions you not to press Shift key for anything so didn't think of pressing it.