Not quite sure what a mongrel is. It seems from the way you're using it, it's what historical linguists call a loanword. I can't think of any words that went from English into Latin and then back into English. When the Romans were in Britain, the tribes who later became the English were still mucking about in Jutland and NW Germany. Latin had pretty much ceased to be a first language by the time that Old English is recorded (e.g., roughly the 7th century CE). The French may be upset by it, but their language has lots of loanwords, besides the base vocabulary from Latin: e.g., a few Gaulish words, Frankish words, Spanish and Italian words, some from Provencal. As for budget, it seems to have come into English via Norman French, which got it from Latin. The Romans had borrowed a Celtic word; I'm not sure if this word was from Gaulish or some other Continental Celtic.
