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According to the UK Government's travel advice, a visitor to Hungary should carry their passport with them at all times.
If I go for a run outdoors in Budapest, will my (smaller, more waterproof and more robust) UK driving licence do instead or will the police cart me off for questioning if they stop me?
Theoretically, you need a passport, but it's very unlikely that anyone will ask for it.
Hungarian law says (10. §)
A külföldi a magyarországi tartózkodása alatt köteles személyazonosító okmányait, illetve a tartózkodása jogszerűségét igazoló okmányokat a törvényben erre feljogosított hatóságoknak ellenőrzés céljából átadni.
which means (my translation, so not legal text)
A foreign person during their stay in Hungary is obligated to present their identification documents and documents supporting the lawfulness of their presence to the entitled authorities for verification.
According to the relevant page (couldn't find it in English) of the Consular Services, you need a passport or identity card for traveling to Hungary, and since there's no ID card in the UK, that leaves only your passport as "documents supporting the lawfulness of your presence", so you must have it on you.
On the other hand, unless you walk around shady places at night alone or with a visibly drunk company, police won't care about you. (I live in Budapest and I can't think right now of anyone I know that was asked by police to identify themselves).
There is a wonderful 5km jogging track in Budapest, around Margit Island, called Hajós Alfréd sétány. Check it out on Google Street View. If you want to change and shower, you can buy a one-day pass to the adjacent athletics club, which also has a 400m track. Google Margitszigeti Atlétikai Centrum. They can also stash your valuables for you at reception, so you can give them your passport for safekeeping if you are worried about being caught without it.
The police won't hassle you if you don't make a nuisance of yourself. I have lived in Budapest on and off for ten years or more, and I often go out without my passport.
