It's 3 months and 28 days - so you would be fine to enter on April 29th.
Even though you have to apply for FLR(M) next, before ILR, you might still need to enter within 3 months and 28 days in order to qualify for ILR after the FLR(M).
If you leave it more than 3 months and 28 days, then your next FLR(M) visa may expire before you have been in the UK for the required 5 years for ILR... so you would need to pay for a second FLR(M) to meet the requirement and then ILR.
If your visa start date is January 22nd, then you have until May 20th 2014 to enter on your visa and still be fine for ILR at the end of your FLR(M) visa.
Of course it still depends on things like the processing time for your FLR(M) and how they work out the 5-year requirement (i.e. if it takes a few months to get FLR(M), then when does the 5 years finish?), but to be safe, I would still try to enter by May 20th.