If you are going to solarize, you don't need to mow or till. Mowing will expose you to the poison ivy.

English Ivy has a waxy coating and is difficult to kill with Round Up. You can try mixing will some detergent to help get through the coating.

If you want a double whammy, I would suggest Round Up, wait 10 days, Round Up again on anything that is still green. Wait 7 days for 2cd Round Up to work. Then solarize.

A good surfactant will make all the difference in how effective the herbicide will be. Wait more than 10 days tho-woody plants will uptake slower so at least 2 weeks and maybe more before spraying again.
Personally, I'd use a a brush be gone product rather than roundup for either ivy. So not cut beore spraying if doing a foliar application. Fall will be the most effective time as plants are drawing nutrients down in preperation for the winter.

What exactly is solarizing?

I think you would need to use black plastic at this time of the year, because the soil will never get to the temperatures needed for real solarization. I'd be afraid that you would just create a green house! Next spring, as temperatures heat up, you could change it out to clear plastic, seal it up properly, and steam the heck out of everything!

Does that make sense? Proper solarization requires sunlight, warm air temperatures, moisture in the soil, and a tight seal all the way around.