Hi. If on grass I would apply the lime first. Sorting the pH first will mean the fertiliser will work better when you apply it. I would suggest waiting for a shower of rain to wash the lime in and then applying the fertiliser about a week later. Bear in mind that a third of the lime will work in yr 1 and a third in yr 2 etc so if pH is very low (very acidic) then it might take more than 1 year to improve.
The calcium in the lime will also help make Phosphate and Potash more available as well as help with the biological activity within the soil.
Did you check your Phosphate, Pottasium and Magnesium levels at the same time (ie a standard soil test or did you use a home pH tester)?