Most of what I know about coronavirus I've learned through DD, so I'm sure there are many here more knowledgeable than me! But I'll try to provide an answer of sorts to what I think you're asking, and perhaps others can correct it.
A test for B memory cells wouldn't necessarily get you the information you want, because, for example, one could have B memory cells that are physically present, but that are not functioning properly to produce sufficient antibodies. What you really want is not to know your level of B cells or any of the other cells and molecules in the fascinating poster that Eru Ilúvatar posted, but rather to know whether you're immune to a second infection of coronavirus. And rather than answering this by dissecting and analyzing all the components of your own immune system, it makes more sense to answer this for the general case: Do humans develop immunity to this coronavirus after a first infection?
Answering this would take a long time, as the doctors have told you. If you want to know whether people are immune for two months or 6 months or 1 year or 2 years after their first infection, then obviously you have to wait two months or 6 months or 1 year or 2 years to get your answer. There are two ways to study this:
1. Experimental: Take a group of people who have recovered from Covid-19, infect them with the virus, and watch them over the next two weeks to see whether or not they get sick with Covid-19 a second time.
This kind of deliberate infection research has been done in the past, most famously when Nazi doctors infected people with tuberculosis or typhoid, and took careful scientific notes as they watched the infected people get sicker and sicker and finally die. The post-war trial that disclosed this research horrified the world, not because it was worse than other Nazi atrocities, but because it was conducted by doctors who had taken an oath to use their skills in healing. In response, international and national guidelines were composed that now regulate research on human health.
To do this sort of experiment today, researchers would first need to decide on the exact details of the protocol they will use, including:
What people will you test? Males, females, or both? Young, old, or both? Those who had severe symptoms, mild symptoms, no symptoms, or all of these? Those who were treated with particular medications? Those who had tested positive for the virus only, for the antibody only, or both? Etc.
How will you infect them? Inject them with the virus, or spray it on their face with a sneeze-machine?
Which virus will you use? Several mutated versions are known to exist already.
What dose of the virus will you use? If nobody gets sick, you won't know whether that's because they're immune, or whether you used too small of a dose to cause disease. If you use too large a dose, and people are not sufficiently immune, you might cause an especially severe disease response.
If people DO get sick after this second infection, how will you treat them? There are ethical issues in infecting someone with a potentially fatal disease before we have clearly effective treatments.
How will you obtain informed consent? This means that anyone who participates in the experiment has to agree that they are doing so voluntarily, and have been informed of what might happen to them as a result of this participation. For example, they might have a slight reaction at the site of the injection, but not get sick beyond that. That of course is what we're hoping for, and it certainly would be great if that's the result. It's also possible that immunity does not develop, and the people will develop Covid-19 a second time. That would be annoying to go through that again, but at least people know what they're getting into. But there's a third possibility, which is that following the second infection they will have a worse reaction than the first time. Perhaps their immune system was partially weakened the first time, and now they are less able to cope and develop a more severe case. Perhaps they had slight unnoticeable changes to the blood vessels the first time, and with the second infection, they are more likely to develop clotting disorders. As far as I know, these last possibilities just exist in my imagination, but the point is that the researchers would have to consider all the things that might go wrong, and notify the participants in the experiment of these in advance.
After the researchers have figured out all these and other details, they would have to send this protocol to a review board, which would discuss whether the experiment, as proposed, is ethical and whether it is likely to lead to significant findings. They might ask for changes, or they might turn it down entirely. Or they might approve it, at which time the researchers would start looking for people willing to be subjects in the experiment.
There are probably researchers considering this sort of experiment, but the process above could take weeks or months, so the rest of us won't hear about it until they announce that they're looking for subjects.
2. Observational. The second way to determine immunity is to simply wait for people who've recovered from Covid-19 to leave home and lead more normal lives, during which time some of them will inevitable come in contact with the virus again. People who get sick with Covid-19 will be asked about their previous infections and previous positive tests for the virus or antibodies, and researchers will be able to analyze how many, if any, of the new Covid-19 cases are in people who had an earlier confirmed Covid-19 infection. If no new infections occur in those who've had confirmed Covid-19 in the past, then we can assume that a first infection causes immunity.
Again, if those with previous infections don't get sick for two months, we'll only know that immunity lasts two months. It will take longer to find out whether this immunity is longer-lasting.
This study will take longer than the previous one, because you have to wait for people to incidentally meet up with a virus somewhere, rather than infecting them deliberately all at once, but it can be started more quickly. In fact, it's started already. We've heard of a few cases in China and South Korea that suggested that a recovered patient had gotten sick a second time, and researchers there are trying to figure out exactly what that means. Is this really a second infection, or a continuation of the first? Was the first infection really the flu? There will undoubtedly be more such cases in the US as people increase their social activity.