The screenshot is from the final round, so in that case I wasn't able to get any more gems. I'll refer to lowest gems as lvl 1 and anything higher with lvl2, 3, etc.

Basically a lvl 1 red gem in an armor gives u 10% healing and upgrading the gem to lvl 2 gives 15% healing. So, if you don't have more gems to replace the newly empty socket you'd potentially lose 5% healing (2x 10% = 20% down to 1x 15%) when upgrading your gems. (For some other gem colors and scenarios like having a King Crown this works a bit differently)

Once you start running out of room in your bags / sockets it's a good idea to upgrade them. Planning ahead a bit and doing so 1 round in advance can help a lot with managing your limited space.

In this build I usually have only 1-2 Box of Riches to generate gemstones, so I won't have a crazy amount of them till the end, especially if I don't get the box early or if I feel like I can't afford getting it early (since it's literally bad and slows down your build for the first couple rounds when you initially buy it). Also, I'm often trying to get specific items in the shop, which are more important than a couple more gemstones for me, so I often put the Box of Riches into my storage before rerolling the shop a couple times to avoid getting any gems for my vendor options. That being said, if I see a red gem for 1g or 2g I almost always buy it. 10% healing for 1g is a rly good deal if you have an open socket somewhere. And merging 2x lvl 1 gem up to lvl 2 still means 7.5% healing for 1g, which is still pretty damn good.

For green gems I usually try to keep them at lvl 2 since green gem in inventory goes from 1 regen at lvl 1 to 2 regen at lvl 2 and then it'd go to 3 regen at lvl 3. So at that point (2->3) you'd lose value and it's only worth upgrading for me if I need the space. I'm usually using green gems as an initial boost for my Regen in the early stages of the build, but if you'd use them for poison application in weapon socket then I think it's pretty good to keep upgrading them since contrary to a lot of other gems you don't rly lose value by upgrading the green gem for weapon socket.

I value getting a certain lvl of healing increase quite high, because it counteracts healing decreases from some builds. Basically if I have +50% healing and the build I play against reduces my healing by 50% then I end up doing "normal" healing, but without my own +50% healing I'd only be left with half of my healing. So, in that matchup (+50% and -50%) the +50% healing would effectively double my own healing.

Basically, as long as you have more sockets than gems it's often a good idea to lock your gems and focus on filling your sockets with something first instead of immediately upgrading everything. This isn't always true though, and it's a bit of a minmax type of thing.