Can I bring a WWII Japanese bayonet I want to buy in the US into the UK in my luggage? I can buy the same knife in the UK but at twice the price?
You absolutely won't be allowed to board a flight with a bayonet in your hand luggage. It would have to travel in checked baggage, but...
There is a lengthy list of prohibited bladed weapons on the UK Government web site which ends in this paragraph:
Offensive weapons which are designed to kill or inflict serious injury and do not have a legitimate use are banned or restricted from being imported into the UK.
Although a bayonet might not fit the description of any items on the list, it certainly comes under this catch-all. I suspect that a customs officer has discretion to confiscate it in any case.
You might be able to import it as an antique if you obtain a relevant import licence and have the item shipped. Doing so would be off-topic here.