I don't know about your phone specifically. My phone when the battery started going dead it would go into a loop and not boot until it was plugged into the charger. With it plugged in, i retrieved most of my info. So it is possible with a good battery or sufficient power it may boot. Also, my understanding is a lot of stuff is saved to phones internal memory unless you specify it to be saved to the micro SD card. If you have an SD card in the phone (not the SIM card); that SD card can be pulled out and read into a desktop computer with an appropriate adapter. I did this with a phone that was lost in a mudhole for a couple days.

You may be more successful swapping batteries than cards.

I don't know what model you have but you may try this as a last ditch effort (Read carefully and choose safe option).  I don't know if this applies to your phone.

https://www.lg.com/us/mobile-phones/VS985/Userguide/097.html