First I have a couple of questions, and suggestions for ways to try to avoid self-funding your PhD:

Do you already have a Masters? If not, it could be a better idea to self-fund a one-year research masters and then try for funding for a PhD after that. That would not leave a 'gap' in your CV, except that probably it's actually too late already to apply for next year.

Have you actually applied anywhere else than Oxford and Cambridge for a PhD? If not, that may be the reason you were unable to obtain funding. There are a lot of other very good universities in the UK (which specific ones are the best will depend on your subject), and doing a PhD with funding at one of those would surely be better than doing one without funding at Oxford or Cambridge. Or if you were not set on doing your PhD in the UK, you could also choose a country (for example the US or Canada) where PhD students are offered funding as a matter of course.
By the way, if you only applied to Oxford and Cambridge, then Noah Snyder's point, which I think is a good one, doesn't really apply, since it's not the case that you couldn't find anyone willing to fund you, it's just that you didn't try enough possibilities.

It may be possible to apply for funding again once you have already commenced your PhD. I know a few people who self-funded their first year and managed to obtain funding for the rest. At least one of them is now a lecturer.

In my opinion you should only try this if you can afford to fund the whole PhD (i.e. don't take out a loan for it!), and even then only if you are very passionate about your subject.

Now to my opinion on your actual question:

You needn't mention how your PhD was funded in job applications, but I suppose that in applications for academic jobs in the UK, employers might notice if no funding source is mentioned. I highly doubt this would count against you at all, as what is really important for your first job after your PhD is the work you have done in your PhD (although an ability to attract funding is highly valued in academia!).

First I have a couple of questions, and suggestions for ways to try to avoid self-funding your PhD:

Do you already have a Masters? If not, it could be a better idea to self-fund a one-year research masters and then try for funding for a PhD after that. That would not leave a 'gap' in your CV, except that probably it's actually too late already to apply for next year.

Have you actually applied anywhere else than Oxford and Cambridge for a PhD? If not, that may be the reason you were unable to obtain funding. There are a lot of other very good universities in the UK (which specific ones are the best will depend on your subject), and doing a PhD with funding at one of those would surely be better than doing one without funding at Oxford or Cambridge. Or if you were not set on doing your PhD in the UK, you could also choose a country (for example the US or Canada) where PhD students are offered funding as a matter of course.
By the way, if you only applied to Oxford and Cambridge, then Noah Snyder's point, which I think is a good one, doesn't really apply, since it's not the case that you couldn't find anyone willing to fund you, it's just that you didn't try enough possibilities.

It may be possible to apply for funding again once you have already commenced your PhD. I know a few people who self-funded their first year and managed to obtain funding for the rest (these were all in the UK). At least one of them is now a lecturer.

In my opinion you should only try this if you can afford to fund the whole PhD (i.e. don't take out a loan for it!), and even then only if you are very passionate about your subject.

Now to my opinion on your actual question:

You needn't mention how your PhD was funded in job applications, but I suppose that in applications for academic jobs in the UK, employers might notice if no funding source is mentioned. I highly doubt this would count against you at all, as what is really important for your first job after your PhD is the work you have done in your PhD (although an ability to attract funding is highly valued in academia!).