How should I route the front derailleur cable on a bike without lugs or guides?
Either Adam's suggestion or this:



If you're going with a very recent FD, like all the Shimano toggle ones and many mountain FDs in general, a lot of them have integrated stops also, so you could do the same sort of routing. Some not super recent mountain FDs have stops for full housing too. If it would play nice with whatever you're building the bike back up with, one of those options plus a bolt-on housing stop as pictured is the cleanest-looking match with what you've got, although it does have the downside that upwards opening housings are bad news in general because they invite water contamination. Most functional would be adding a nylon BB guide as you say, although not very aesthetic since having a top mounted chainstay stop marries you to the RD cable guide you've got.

I believe there have been a handful of old front derailleurs with weirdo integrated cable guides that reach all the way down. To my awareness these are all super low end, and I wasn't able to find a picture. I don't know much about Dawes but I doubt this what it came with, and it's likely not worth pursuing now.