The civil registration of Birth Death and Marriage began in July 1837 and has always been pretty much in the same format.
Local register offices can provide a quick chesk for someone registered in their district now using a cpumpterised index.
The general Register Office index has three parts, the birtths, deaths and marriages. If you think you have his name, and you have an idea where he may have lived, the it would be quicker to visit your local register office possibly.
If you can't do that, you can go into any local library and search the civil regsitration index using Ancestry.co.uk Library edition. the GRO Index for all types of entry has each year split into 4 quarters, Jan -March, April - June, July - Sept and Oct - Dec.
You need to check each suname in each quarter if you are unsure of the dates, but on some sites you can really specify more info to give you less results to analyse.
As long as you start your research by compiling as much info as possible about Grandad Palmer, you should be able to eliminate the wrong ones and find the right one. I suggest you start writing down details of any siblings of your Mum and any known aunts uncles to provide a time line, speek to your relatives who may be sitting on family documents that would give you a head start.
If your Mum has any ideas on where your grandad may have been from it would help, he may have had an accent for instance.
As for war record, if it is WW1 it shouldn;t be too hard, if it is WW2 some are still closed but there is alot of info in other threads on this topic.
Good Luck