Do UK embassy in Nigeria carry out physical address verification on their visa applicant's home and office address.
It is likely that they do not check each and every address on every form send in to them. It is more likely that they check out a few at random. How few or many will be based on the results of those checks as well as on manpower available to them.
But stick to the basic rule, never lie to immigration and visa officials. It is not about the chance to be caught on a lie, it is the results when caught.
When caught on providing wrong information the usual reaction is a ban, at least for the country you applied for but often also countries they work together with.
At the least you will be denied the visa and will have to fill out that fact whenever asked about such.
If you do not want the people where you live or work to know about the application, it is rather unlikely that the people checking will tell why they check, but it is not impossible. You could try a request not to disclose the reason they check, with an explanation why, but I do not know whether that will work and whether that will be a negative in the visa process.