In answering this question, we have defined 'the UK's overseas military bases' to mean the Permanent Joint Overseas Bases:
BRITISH FORCES SOUTH ATLANTIC ISLANDS (BFSAI)
The defence and security of the Falkland Islands remains a HMG priority, and as such we undertake regular assessments of potential military threats to ensure that we retain an appropriate level of defensive capability. The Ascension Island airfield is critical to our ability to support the Falkland Islands.
BRITISH DEFENCE SINGAPORE SUPPORT UNIT (BDSSU)
The UK maintains a permanent military presence in Singapore as a commitment to the Five Powers Defence Agreement. In addition, on request, BDSSU supply fuel to other nations' warships. The small team build on several strategic relationships in South East Asia and with the US contingent, who are collocated in Sembawang.
British Forces Cyprus (BFC) protects and delivers against the UK's strategic interests, which includes its current contribution to both the air and land elements of OP SHADER, the UK commitment to the counter-Daesh coalition. RAF Akrotiri enables the rapid deployment of military capability within the region and beyond.
The Sovereign Base Areas (SBAs) in Cyprus is the only UK Overseas Territory to be Administered by the Ministry of Defence. The UK maintains sovereignty over the SBAs for military purposes, whilst protecting the interests of those who reside or who work in the Areas and administering them in cooperation with the Republic of Cyprus.
Diego Garcia, the main island and military base in the British Indian Ocean Territory (BIOT) under UK sovereignty, remains an important base for the defence needs of the UK and US. The Small UK Military contingent work with the BIOT Administration to demonstrate sovereignty and protect the 64,000km2 of pristine, ecologically important waters from illegal fishing activity.
Duqm is a developing port situated approximately 500km south of Muscat. Important to the economic development of Oman, its location makes it strategically useful for the UK, US and others: it will facilitate maritime basing east of Suez but outside of the Gulf, including a dry dock capability able to accommodate submarines and QEII Class aircraft carriers.
The UK maintains a permanent military presence in Gibraltar, maintaining key sites and Air and Sea Ports of Disembarkation, which allows reinforcement to ensure Gibraltar's security and territorial integrity. UK sovereignty of Gibraltar is demonstrated through a combination of presence and posture in order to reassure the Gibraltarians and to enable the maintenance of a Forward Mounting Base, and to provide security to one of the UKs overseas territories.