I assume that you or a partner are going to work there. Otherwise Cyprus uses the same rules as the rest of Europe ie if you're inactive you have no entitlement to health care above that of the EHIC card which is only for short term visits. If not then you would have to use private insurance which would be unavailable for someone with T1. If you work then you would have the same entitlement as someone working there.
If you are looking for work then the UK will pay for your affiliation to the other countries scheme. You will need a form S1. This lasts for up to 2.5 years (depends on time on year you apply and contribution record) ....... I assume that with a 7 year old you or your partner aren't in receipt of a state pension when the SI is permanent.
One thing I would say is not to chance going without health care sorted out both for you and your daughter. People in the UK often don't realise that else where, not only is healthcare not free in the same sense as in the UK but it can be very expensive. It's not unknown for people who have moved here to France without medical cover to have to sell their homes to pay for medical care.
I found this thread on an expat forum. You really need to investigate all sources of info.http://www.expatforum.com/expats/cyprus ... yprus.html