Sorry for your loss.
1. The first place to check is the Royal Mail Pensions Service Centre on 0114 241 4545. They should be able to advise if there was anything in your father’s pension, including if there was a payout of this type. The pension schemes changed several times over the years, so I wouldn’t like to say what was even in the scheme from that time.
2. We used to have death in service benefit, but obviously this wouldn’t apply here.
3. If your Mother is struggling financially then it might be worth applying to the Rowland Hill Fund. I am not sure if your Mum qualifies but she could ask. There are similar funds for other some other employers so it might be worth going through the employment history of both parents and seeing if you can find something.http://www.rowlandhillfund.org/general/home/