Linux

While trying to test this on my system, I came across a few issues. I found a more modern and standardized way to check for login after sleep, and have updated my previous comment with a new script. I have new instructions though that are more comprehensive that you can follow.

Create 2 files bluetooth-reconnect.sh and bluetooth-reconnect.service. Using your favorite text editor, edit these files so they will contain the following:

bluetooth-reconnect.sh

#!/bin/bash

gdbus monitor -y -d org.freedesktop.login1 |
  (while read x; do
    if echo "$x" | grep -q "{'LockedHint': <false>}"; then
      bluetoothctl connect A1:11:22:3A:CD:F1
    fi
  done)

bluetooth-reconnect.service

[Unit]
Description=Reconnect Bluetooth after waking from sleep
After=default.target

[Service]
Type=simple
ExecStart=/usr/local/bin/bluetooth-reconnect.sh

[Install]
WantedBy=multi-user.target

Open a terminal in whatever location you created them (you can right click in your file manager and open in terminal, or use cd to navigate). Now move them to the correct locations (you will need sudo privilege for this):

sudo mv bluetooth-reconnect.sh /usr/local/bin/bluetooth-reconnect.sh
sudo mv bluetooth-reconnect.service /etc/systemd/system/bluetooth-reconnect.service

Make the script executable:

sudo chmod +x /usr/local/bin/bluetooth-reconnect.sh

Enable and start the service:

sudo systemctl enable bluetooth-reconnect.service
sudo systemctl start bluetooth-reconnect.service

Check to make sure the service started correctly (the "Active:" field should say "active (running)" in green).

sudo systemctl status bluetooth-reconnect.service

This should now do everything automatically. This has been tested and is working on my Fedora Workstation system (uses GNOME). This should be distro independent, unlike my previous answer (and also without the syntax error I had in my initial submission).

To uninstall:

sudo systemctl stop bluetooth-reconnect.service
sudo systemctl disable bluetooth-reconnect.service
sudo rm /etc/systemd/system/bluetooth-reconnect.service
sudo rm /usr/local/bin/bluetooth-reconnect.sh