One important thing you need to know about the German Doktorgrad is (alongside other non-academic titles) that its use is restricted by federal law. Adding a Doktortitel to your name in public without being entitled to do so is a punishable offence. This means that you should not trust any answer that you get from the Internet on this topic. Make sure that you verify whatever answer you get with an official side. One way of doing so is pointed out in @Cachapa's useful answer from the point of view of someone who already went through this.
Having said that: First of all, a British Doctor of Philosophy is indeed considered equivalent to a German Doktorgrad. You can verify that using this database (unfortunately only available in German). This means that you're fully qualified for whatever position or task that requires a Dr.
However, while you're entitled to use your PhD in public as an academic title, it is not possible to convert your PhD into a German Doktorgrad, and doing so might be an offense according to German law. This is based on a statement issued by the Kultusministerkonferenz (the "Standing Conference of the Ministers of Education and Cultural Affairs of the Länder in the Federal Republic of Germany"). These are the crucial points:
- Akademische Grade sind in Deutschland rechtlich geschützt.
- Der ausländische Grad muss ordnungsgemäß verliehen sein.
- Die Umwandlung eines ausländischen Grades in einen deutschen Grad ist nicht möglich (…).
- Academic titles are protected by the law in Germany.
- A foreign title has to have been granted lawfully.
- It is not possible to convert a foreign title into a German title.
This statement is very explicit in my opinion, and it means that you cannot just add Dr.-Ing. to your name if you haven't been granted the title by a German university. It also means that no German university can grant you a Dr.-Ing. simply because you have a foreign equivalent title.
Yet, you can use your PhD in public, because it's acknowledged as an academic title. Plus, the Kultusministerkonferenz has decided that EU members can also choose to use Dr. instead of the designation used in their country of origin, this is regulated in (2) in the "Vereinbarung
über begünstigende Regelungen der 'Grundsätze für die Regelung der Führung ausländischer Hochschulgrade'" (thanks to @Pont for pointing that out), but this does not include the "-Ing." part. To quote:
- Hochschulgrade aus Mitgliedstaaten der Europäischen Union (EU) oder des Europäischen Wirtschaftsraumes (EWR) […] können in der Originalform ohne Herkunftsbezeichnung geführt werden.
This means that academic titles from the EU, the EEA may be used as part of the name in their original form without stating their origin.
- Inhaber von in einem wissenschaftlichen Promotionsverfahren erworbenen Doktorgraden, die in den in Ziff. 1 bezeichneten Staaten oder Institutionen erworben wurden, können anstelle der im Herkunftsland zugelassenen oder nachweislich allgemein üblichen Abkürzung […] wahlweise die Abkürzung „Dr.“ ohne fachlichen Zusatz und ohne Herkunftsbezeichnung führen. […] Führung beider Abkürzungen ist nicht zulässig.
This means that a holder of a doctorate from one of the countries described in (1) may decide to use the abbreviation Dr. instead of the abbreviation that is usually used in their country of origin, but without the addition of an abbreviation indicating the title-granting discipline – that's the part referring to the -Ing. suffix. Also, it's not possible to combine both abbreviations, i.e. "Dr. Schmuddi, PhD" is not allowed.
Edit (March 2021)
The answer above reflects the state of affairs from 2017, a time at which the UK had already voted in favor of Brexit but was still a EU member. Now that the UK is not a member state anymore, paragraphs (1) and (2) don't apply to UK PhD holders. Consequently, the "Vereinbarung" was revised in 2019 by adding the UK to the list of exceptions in paragraph (4) (alongside Australia, Israel, Japan, Canada, and the US):
- Inhaber von folgenden Doktorgraden [...] Vereinigtes Königreich: „Doctor of ...“ mit jeweils unterschiedlicher Abkürzung [...] können anstelle der im Herkunftsland zugelassenen oder nachweislich allgemein üblichen Abkürzungen die Abkürzung „Dr.“ jeweils ohne fachlichen Zusatz und Herkunftsbezeichnung führen.
According to this revision, any scholar from the UK who holds a PhD (which is an instance of a „Doctor of ...“ degree) may still use Dr. in their name, but as before, you may not add the degree-granting discipline. So, at least as far as the regulations concerning using an UK PhD in Germany is concerned, Brexit didn't cause any substantial change after all.