The law in the United Kingdom states a person is a UK dual national if he/she holds multiple citizenships. Normally, there isn't a restriction on nationals from Britain having dual citizenship elsewhere.
If you are a British national and you acquire another nationality, normally you will not lose your British citizenship. However, there are many things to consider beforehand and there are some existing restrictions and regulations to this law.
When considering dual citizenship, there are several countries that restrict or don't allow dual nationality at all. Be aware if you are a national of a country with restrictions and decide to become a British national, the law of your country can consider your nationality voided or even refuse to recognize your new nationality in Britain.
If you are concerned you may lose your existing citizenship when applying for British nationality, contact the Consulate or High Commission of the country whose nationality you already hold for further advice before applying.
However, be ready to renounce your British nationality if you are British and the country does not allow dual nationality. Also, if you have been a British national and the citizen of another country since birth, you may be required to renounce your British nationality in order to keep your other citizenship.
A State has the right to not give consular assistance to a national if they are in a country the national has dual citizenship in. For example, if you are British and Chinese, the British representative in China cannot give you consular assistance.
Because of this, if you are planning to visit your former country and not sure if you lost your nationality, ask the country's Consulate in the United Kingdom. Alternatively, if you have not lost your old nationality but want to give it up, they will also be able to assist you.
If you obtain written confirmation that you gave up your former nationality or no longer have it, or obtain a renunciation certificate before traveling abroad, this can grant you the same consular assistance which is provided to all British nationals in your former country.
Also take into consideration a married person automatically has their spouse's nationality and their children have a parent's nationality wherever they were born under many countries' international laws. Please keep this in mind when traveling, as this could also apply to your wife, husband, or children if they are traveling with you, as well.
It should also be noted British citizens, as well as certain Commonwealth citizens and British subjects, have the right of abode in the United Kingdom, which means they can live and work in the United Kingdom without any restrictions.
However, since December 21 2006, a certificate of entitlement to the right of abode can't be issued and any recent certificate of entitlement may be canceled if the person has a passport (or Identity Card) which describes them as a British citizen or British subject with the right of abode. All of this should be taken into consideration for UK dual nationality both for yourself and your family.