This link might help

http://alien.slackbook.org/dokuwiki/doku.php?id=slackwarexe



for gentoo it might be a bit easier. Well easier if you have done gentoo before.

No gentoo experience, I'm afraid.
I'm left with questions still though. Bad habit of mine, I guess.
That's written with Slackware installing from Slackware in mind, basically.
Ubuntu's support documentation, similarly, has a method that naturally assumes Ubuntu from Ubuntu.
Now I'm left wondering how it works for Slackware from Ubuntu.

Not at all -- you are contemplating a complex procedure in difficult circumstances so better to lean on LQ community knowledge, prepare as best you can and go in with eyes wide open. The better you understand the process before starting the more smoothly it will go and the better able you will be to workaround anything unexpected.

ubuntu on your laptop doesn't get a look-in in all this; your computer's BIOS will start a PXE boot which will lead into the Slackware installation procedure. Early on in the procedure you will format the HDD and blow away the existing ubuntu installation -- along with all your data so (you already know this!) make sure you've backed up everything you need to another computer first and you've got nothing to loose except time.

Regards setting up BOOTP, TFTP, NFS (or whatever) on another computer, the instructions are for Slackware but, deep down at that level, most Linux systems (including ubuntu) are similar and the instructions should (ha!) not need much adaptation to suit.

Ah, you misinterpret what I meant - I'm trying to install Slackware, with the computer providing it using Ubuntu - the Ubuntu guide assumes you're installing Ubuntu on the other computer, and the Slackware one linked to seems to assume you're using Slackware on the providing computer.
I realize that there's not much difference at the basic level, but in all honesty, reading through the guide on how to do it kind of makes me feel a little intimidated. I don't mind using a CLI once in a while, when necessary, but this needs it a lot more than I'm used to, and with commands I've never used before, which bothers me.

Understandably!

If you are really not comfortable with the CLI then Slackware may not be the best solution for you (it does require CLI work to set up) but it is fast and stable ...

The hardware-based solutions discussed earlier in the thread would be easier than installation over the network and would leave you with a more flexible machine afterwards ...

Installation from local HDD is simpler to set up than installation over the network but requires finding a way to boot a file system management utility over the network (or from HDD partition) to free up some space ...

So many choices, and none of them easy!

Well, I'll have to get down and dirty with it sometime if I'm going to get my old laptop working, and knowing at least the basics of how to net-boot from those two guides is a start.

As to which distro to use, I've had all kinds of suggestions and choices - the Zegenie studios chooser suggested, in order, Fedora, Arch, OpenSuSE, Slackware and Gentoo.
Gentoo and Arch, given that they're aimed at more experienced users, I think I can rule out.
The rest are all options really, though it was suggested earlier in the thread to use Slax, which I've looked at and the modular custom liveCD is intriguing - though there are some oddities (Why does pygame ask for google earth, for example?)

So far, Slax is the one I'm looking at trying - given that it's based on Slackware, and what you've said, it suggests a CLI installer, which I'll at least try first.

The problem with the hardware solutions though is that firstly, the adapter I need, I don't have the money to get right now, but might after christmas - but I'd prefer to have my laptop ready before then, really.
And a local HD install, well, I'd like to - but it's really a case of completely repartitioning in advance.

So at the moment, what I think I'd like to try, is Slax from the LiveCD, over the network... probably not the best choice, but if all options aren't easy, then it's a case of what can I do?

Right!

Thanks to help from all you nice people, some of Ubuntu's own support on their forums and IRC, I have now turned the Swap partition into a separate one, and I think there's some progress.
Since I don't want to cross post - there's a thread in the Distributions forum (Not the subforums, since there's no Slax one) that has the next problem I've been having, with Grub. I don't know if any of you can help with that or not, I'm hoping someone can, because I think it might well be the last problem on the way to installing it.

Thanks again to all of you!