A bit more specifics about your process may help an expert give you more help with the problem you're having. Some thoughts:
My group is currently working on a Biogas Digester. Although, after 2 weeks, no methane was produced. We have some problems that we need answers/opinions. Here are they:
1. How long will it take for a 20 gallons mixture of cow manure and water (1:1) to produce methane?
I don't think you can expect a biological process, applied to a biological source, to have a rigidly defined time frame. How ever, in the two weeks, what has happened? Is the mixture fermenting? Is it releasing gas? What is the compositions of that gas, and how do you know the composition? This helps us not only know whats wrong, but also your technical level -- are you part of a small group working on a process, or a home scientist?
2. Is there any way for us to fasten the process? Such as, can we use a catalyst (yeast) to fasten the process up?
No. I suspect that won't work at all. Why do you think it might? Why do you call a living thing a catalyst? Do you have a resource that suggests that yeast produces methane?
3. What can we do if our digester still doesn't methane? Do we need to replace our slurry (manure and water mixture) with new one or is there any remedy that we can do to still use the first mixture that we made?
Your question leads to a bunch of off topic questions of mine: How much slurry do you have? Where did it come from? How did you get it, and how will you dispose of it? How will you get more, and what will you do with the excess? When you've made all the methane you can, how will you dispose of your "spent" slurry? We need this information to gauge your technical level, so we know what help to give you.
Briefly, I know very little about methane production. Back in the 1970's I read a children's book on planning an experiment to produce methane. Basically, a digester contained a slurry of manure: chicken litter was best, cow manure next best, and horse manure least yield of all. Existing fecal microbes should provide the enzymes needed to produce methane, if the digester has no oxygen. Carbon dioxide is produced to start with, until oxygen is depeleted. So the produced methane is always contaminated with other gasses. After all the methane is spent, a small amount of slurry can be used as fertilizer for home use.
But again, this is a children's book explanation, you may need something better. On the other hand, you seem to lack the very basics of methane production, analysis, waste handling and you've given us no idea of the yields you expect. So I'm left wondering what help you really need.