Time and patience are the only solutions I know of, I'm afraid.
My last (of 4) dgc was the worst for this. My dil would dearly love to have left him with me, as he has never been a good sleeper either day or night, so she really needed the occasional rest.
The best we could manage for many months was for dil to (breast) feed him and then for me to immediately take him out for a walk in his pram. There would be some crying while we were out, but not usually persistent enough for me to have to take him back immediately. We were rarely out more than 45 mins, though.
Several months before going back to work full time when dgc was 12 months, dil had to start going in to the office for quite a few 'keeping in touch' sessions, which meant that she would have no choice other than to leave dgs with me for several hours. I think he was about 8 months old when this started, and it was a very difficult time. The only time he didn't scream non-stop was if I walked around holding him - if I tried to so much as sit down with him, he was distraught.
Plus it was impossible to put him down for a nap in the cot (even his own dad couldnt do that - it took mummy and the boob!), so the only way I could get him to sleep was to bundle him crying into the pram and walk up and down the street until the movement and exhaustion finally got the better of him and he would sleep for a short time.
All I can say is that it got better gradually. By 12 or 13 months I could transfer him from buggy to the cot without waking him. Around then, he started letting me sit down with him too, rather than having to stand all the time, and he started being happy to see me when he was dropped off in the morning.
Nursery 3 days a week started at 14 months (the other 2 days with me), and although that was predictably hard for him to begin with, he settled there after a few weeks. Just to add that it's true what they say about nursery 'magic' regarding getting little ones to sleep in a cot - the very first week there they succeeded where we had failed, and he brought the magic home with him as it has worked for me too ever since that week.
Now, at 18 months old, he is very happy when left both with me and at nursery. In fact, I was told that he is now trying to comfort the 'new' babies who have just started at nursery, by taking them toys that he thinks they might like when he hears them crying. ☺