in general, mice do not get asthma. However you can induce some features of asthma like eosinophilia or AHR. This requires about 4 weeks in the classical models unsing OVA i.p. sensitization and aerosol challenge.
I agree with Armin - to induce some asthma-like characteristics like BALF eosinophilia, high serum IgE levels, lung inflammation and mucus secretion usally you need several weeks.
After one-three OVA i.p. treatments we usually wait some days (at least 7-10) until you use repetitive aerosol or i.n. challenge to provoke a local response in the lung. The quantity of sensitizations and waiting time afterwards until challenge differs in the literature. Several days after i.p. treatment are required for stimulation of plasma cell development and Ig synthesis to get a nice detectable Ig titer against OVA. The shortest time frame for the whole immunisation procedure I tried so far are 12-14 days in total. However, than some parameters might be not as high, and it only represents an acute phase. To get chonification you have to perform weekly rechallenges over a longer time period (2-3 months). Good luck for your experiments!