So, it seems that only people with older graphics cards have this problem; I checked the minimum system requirements for C&C remastered, and I'm going to assume your CPUs are up to snuff; for I don't think the game would even start, so let's look at the minimum graphics card requirements:
AMD Radeon HD 5570
Nvidia GT 420.
So, let's go with the Nvidia first, the GT 420 has an amount of video memory that 2GB. Looking around for a GT 550, I find one mention where it says the card has a MAXIMUM amount of video memory of 1536MB, or 1.5 GB; aka 0.5 GB less than the card mentioned in the minimum requirements. Which rather fits, because it very much sounds like your cards straight up don't have enough memory to load all the sprites and textures; and so won't display the stuff that gets loaded in last and thus can't be loaded because not enough memory.
As for the Radeon user, you mentioned HD 4850; I'm not familiar with AMDs naming conventions, but usually, a higher number means a newer card, and usually a more powerful car (but not necessarily a more powerful card, depending on what range in the series your card is). So at first glance it seems your Radeon card also doesn't meet the minimum system requirements, and it seems it's also the video memory issue. Check the the amount of video memory on your card vs. the minimum requirement HD 5570 memory and see if yours has less; probably both brands require the same amount of minimum memory, namely 2Gb.
This would also explain why onboard memory will run the game; because from what I gathered most onboard graphics solutions don't have their own video memory, they use system memory for video memory, and if your system memory is big enough to accommodate C&C's program memory requirements and has enough left over to give the onboard system 2GB of video memory, the onboard solution might indeed manage C&C remastered, while your separate graphics cards don't.
If I'm right about this, it's not the developers fault, nor a bug, your graphics cards simply don't meet the stated minimum system requirements.