There is no restriction for a single entry supporting visa to be unused while requesting an eVisa for Turkey. A supporting visa, even if single entry and used is still accepted as long as it is valid by date on the date entry is requested into Turkey.
As long as you enter Turkey within the validity period of your Visa (Even if used) you are still eligible for eVisa.
For example if your UK visa expires on 31st December 2017 and it is single entry and you have already used it you can still enter Turkey on 31st December 2017 and then you can stay in Turkey for as long as your eVisa allows.
This is very clearly stated on the eVisa website.
What are the criteria for the validity of my supporting document (visa or residence permit from Schengen or from US, UK and Ireland)?
The only requirement for your visa/residence permit to be used as a supporting document is that it should be still valid (by date) by the time you enter Turkey. Previously used or unused single-entry visas are accepted as long as their validity date covers your entry date to Turkey. Please note that e-Visas of other countries are not accepted as a supporting document.
For the sake of completeness, you do not even have to be in Transit to use this visa. It is a tourist visa and you can take a direct flight from your home country to Turkey and back if you like. Whether you go before your UK trip (as in the case of OP) or after, it does not matter. You can make it 2 trips equally as well. One to the UK, one to Turkey.