OK, let's set up a specific hypothetical situation so we have some details: A virus (we'll call it Human Nasty Virus 1 (HNV-1)) can infect T cells (a type of white blood cell in the immune system) by binding to a certain receptor on its surface, which we'll call the Nasty Virus Receptor (NVR). Scientists have found that HNV-1 absolutely needs a certain protein sequence in NVR to bind, as people with a certain mutation lacking that sequence are completely immune to HNV-1 infection.
You sign up for a clinical trial using gene editing to specifically remove that DNA sequence from the NVR gene. Once that changed gene is transcribed and translated into protein, it is expressed on the surface of the cell, and HNV-1 can no longer bind.
T cells are derived from stem cells that live in the bone marrow. In order for the gene therapy to work, doctors will need to perform a bone marrow transplant - they will remove a sample of bone marrow, give it to the scientists for the gene editing process, then reinject the edited cells back into you. The editing process involves subjecting purified stem cells to a process called CRISPR, allowing them to grow in culture for a while, then screening the cells to select only those where the editing was successful. The successfully-edited cells are separated, cultured some more to expand their numbers, then given back to the doctors, who put them into your body, either into the bloodstream or directly back into the marrow.
So now you have your edited cells back inside you - how long will it take to become immune to HNV-1? Well, the honest answer is that, in the absence of ablative chemotherapy (see the bone marrow transplant link above for more info) prior to reinjection of the edited stem cells, you may never become completely immune. Here's why: Unless all of the bone marrow stem cells in your body are destroyed before reinjecting the edited ones, you will still have fairly high numbers of what are called "wild-type" or un-edited stem cells and mature T cells in your body, and those stem cells will continue making new wild-type T cells. Additionally, there are already wild-type T cells in circulation and in secondary lymphoid organs such as lymph nodes and the spleen, and some of these cells can be very long-lived (months to years). If for some reason you decide to undergo complete ablative chemotherapy to essentially destroy your immune system, the reinjected stem cells will eventually repopulate everything and you'll be immune to HNV-1, although you will have potentially lost your immunity to everything else you've ever been exposed to, unless the chemo is done in such a way as to preserve your immune memory cells.
This is actually a best-case scenario, as we were working with cells that can (relatively) easily be removed from the body, edited, screened, and reinjected. If you were trying to cure a genetic disease that affected solid tissue like the muscles (see Duchenne muscular dystrophy for an example), you obviously wouldn't be able to remove all the muscles and edit them ex vivo, you'd have to do the editing in the body, most likely using a viral delivery system for the editing machinery. The viruses wouldn't infect every single cell, and the editing wouldn't work in every cell that got infected, so the best you could hope for would be a partial cure. For some diseases, this may be enough, and many companies are working on this from various angles. There are also other options besides CRISPR-mediated gene editing that have higher success rates in vivo.
Finally, getting back to your original question of how long it takes from DNA alteration to expression of the gene product, the answer is that it depends on the gene and the context. In the case of our hypothetical situation earlier, as soon as the NVR gene's DNA is edited, the new sequence begins to be transcribed and translated into protein. There is still some wild-type protein left over initially, so it depends on the transcription rate of the gene and the turnover of the existing protein. Some proteins have an extremely long half-life, on the order of years or decades (or more!), while some degrade within minutes to hours of being synthesized.