No exit stamp on leaving Schengen area from Budapest to UK

In theory, there is indeed a way to fix this.

Once you proved that you did in fact leave the Schengen area on that particular date (presumably by showing your UK entry stamp and/or some other proof of your presence in the UK), the border guard should indicate that in your passport and may give you a special form called “Approval of evidence regarding the respect of the condition of duration of a short-stay in cases where the travel document does not bear an entry or exit stamp”.

At least that's what's supposed to happen according to article 11 of the Schengen Borders Code:

1. If the travel document of a third-country national does not bear an entry stamp, the competent national authorities may presume that the holder does not fulfil, or no longer fulfils, the conditions of duration of stay applicable within the Member State concerned.

2. The presumption referred to in paragraph 1 may be rebutted where the third-country national provides, by any means, credible evidence, such as transport tickets or proof of his or her presence outside the territory of the Member States, that he or she has respected the conditions relating to the duration of a short stay. In such a case:

   (a) where the third-country national is found on the territory of a Member State applying the Schengen acquis in full, the competent authorities shall indicate, in accordance with national law and practice, in his or her travel document the date on which, and the place where, he or she crossed the external border of one of the Member States applying the Schengen acquis in full;

[…]

In addition to the indications referred to in points (a) and (b), a form as shown in Annex VIII may be given to the third-country national.

4. The relevant provisions of paragraph 1 and 2 shall apply mutatis mutandis in the absence of an exit stamp.

The form is reproduced in annex VIII of the same regulation and would say something like:

We, the undersigning authority, XXX have before us […]

Having regard to the evidence relating to the duration of his/her stay on the territory of the Member States that he/she has provided, he/she is considered to have left the territory of the Member State XXX on XXX at XXX at the border crossing point XXX.

The note in your passport or the form can then be used to prove you left the Schengen area on that date in lieu of the exit stamp you should have received at the time. Thus, German (for example) border guards could in effect correct the record on behalf of their Hungarian colleagues.

In practice, I guess you could ask border guards for this next time you cross a border or approach the border police/immigration authorities within a Schengen member state and ask them to annotate your passport. Make sure you have solid evidence that you did leave the Schengen area with you.

Alternatively, getting a new passport should also solve the problem. You might need to explain all this one more time to get a visa in your new passport but after that, the missing exit stamp would not be an issue anymore.