Hello and welcome.
We have experience of bringing up a mixed race grandchild up in Kenya, since she was about 2 years - she is now 9 and overall it has been very positive for her. I can't think of an instance where she has been made to feel out of place and actually (from my own viewpoint) its much less of an issue here, than it is in the UK. She is one of a very few mixed race children at school, which has also never been an issue.
I will warn you that items such as formula milk and nappies (diapers) are pretty expensive here, compared to the UK, so this should be a real concern. When we were buying formula milk it was about Ksh 900 for a smallish tin. I am sure it costs considerably more now, for the 'local' Nestle Nan. If you want imported (Cow & Gate, etc) you can double the price.
Healthcare; There is a decent children's hospital in Muthaiga, Gertrudes - private of course. Unfortunately for the best healthcare you will have to go private and I would also highly recommend heath insurance (check a few websites to find out prices - AAR, Jubilee, etc). The advantage here is that you can walk into a doctors consulting room and be seen almost immediately. Results are back very quickly.........BUT of course you have to pay. Fees really depend upon what the problem is, but expect to pay perhaps Ksh 2000 - 4000 for a consultation (without insurance). Kenyan doctors are fond of prescribing medication. If either of your children have any health issues, then staying in the UK would be advisable, due to the potential costs in Kenya.
Schools: Depends whether you want an international school and UK curriculum, or the Kenyan system. International schools are incredibly expensive compared to Kenyan system schools. The Kenyan education system is pretty good - we swapped our granddaughter to a Kenyan school, from a British system school just over a year ago and don't regret the decision. You could also consider state primary schools, which are the cheapest, but many would argue that they aren't quite as good.
Areas to live; in my experience expats and locals don't mix very much. I don't really think that it matters too much where you live from this respect as your kids can and will mix when you go out and at school. Where we live now, I am the only white person in the area and our granddaughter is the only mixed race child - we don't feel at all out of place (am in a mixed race relationship). Our priority has always been to have somewhere safe and secure.
If you can't afford Westlands, then I feel that many of the better areas of Nairobi would be out of your price range. The majority of europeans would be hesitant to live in cheaper areas, for security reasons. Nairobi is a very expensive city. Is it a must you live in Nairobi? Places out of the city are significantly cheaper.
Work: You will be easily able to reside in Kenya with a Dependents Pass (as would your children. However, this pass assumes that your partner is working and supporting you. You are not permitted to work with this type of pass. One more thing. You are only entitled to a Dependents Pass if you are married. If you are not married then settling in Kenya becomes much more difficult.
To work you must have a work permit. To obtain one, you must have the offer of a job, then your prospective employer, having evidenced that the position cannot be filled by a Kenyan, applies for your work permit.
I have tried to touch on the issues you have raised. Please ask for clarification for anything else.
