Congratulations for your life decision!
Here are the documents needed for marriage at the civil status office in Kazakhstan:
1. Application, found here: http://egov.kz/wps/portal/!ut/p/b0/04_Sj9CPykssy0xPLMnMz0vMAfIjc7PyChKtUvKTS3NT80r0w_Wj9KNgPM8U_cgwQyMDIDA3NTYGSeUk5qWXJqan6kcWleoX5OZalDsqKgIA2pHNTQ!!/#z260 (I am sorry it is only in Kazakh language, your partner should be able to translate it);
2. Certificate of No Impediment for you, taken from Britain (the relevant instructions, depending on your domicile in Britain, are found here: https://www.gov.uk/marriage-abroad/y/kazakhstan/uk/partner_local/opposite_sex);
3. UK passport for you, with the temporary residence authorization (which you can obtain on-line through the procedure described here: http://egov.kz/wps/portal/Content?contentPath=/egovcontent/citizen_migration/citizenship_of_rk/passport/pass008_mvd&lang=en). Your passport will have to have a notarized translation into Kazakh and / or Russian at a local notary in Kazakhstan (same goes for any foreign document);
4. Identity card or passport for your partner;
5. If she is between 15 and 18 years of age, documents proving that a marriage is necessary (for example, if pregnant);
6. Information about previous marriages from both of you (specifically, if there were any marriages, certificate of marriage and decree absolute of divorce, obtained in the UK for you, and in Kazakhstan for her), with apostille in your case;
7. Information about children (if any), specifically birth certificates of any children which either of you might have;
8. Information about health status and financial condition of each other, as well as on absence of impediments of entering into marriage. Specifically, this can be obtained by giving a common declaration at a local Kazakh notary, stating that you have communicated to each other your health status and do not have communicable diseases (it would be best to do a health check locally, and keep the receipt for the civil status office), as well as having communicated your financial status to each other, and that you are not barred from marrying for any reason (i.e. kinship, physical or mental disability etc.).
The ceremony is held one month after depositing all these documents at the civil status office.
The marriage certificate is issued straight away. To use it abroad, you will have to get it apostilled in Kazakhstan. Check the competent authority and its contacts on this page: https://www.hcch.net/en/states/authorities/details3/?aid=368
I hope my answer was useful and look forward to your rating, which is essential to my activity.
Dr I L Vlad