There are two possibilities. Firstly, she may be giving birth at a private clinic. In this case, as far as I'm aware, there is a fee that should be paid at the time of the operation (not sure if you pay first or get an invoice just after). I imagine it'd work much like is does in the UK.
The second possibility if that she's having the baby with the state service, equivalent to the NHS. Just like the NHS, this is 'technically' free. However, the doctors here are underpaid and have no equipment and little medication, so the usual practice is for the patient to 'pay' for the operation. This will including 'tipping' the various people who might take part in the operation; surgeon, midwife, nurse, anaesthesiologist - and the doctor will purchase the consumables he/she needs (such as pins, artificial joints, or whatever the operation requires). They'll also use the money for subscriptions to medical journals and attending symposiums and conferences related to their fields so that they can keep up-to-date with the latest in their specialisation. It also tops up their salary. All of this is illegal/unofficial, of course, but if they couldn't do it, there'd be no doctors in the public sector.
This is where the problem occurs. Most people see this 'payment' as a kind of bribe, and in a lot of cases it is. The problem is knowing which doctor uses this money for the above purposes and which just pockets the whole lot. I know that last year they started cracking down on this and so most decent specialists started preparing to move to other countries where they can earn a reasonable salary without having to beg the patients for money, which could lead to a massive talent drain in Romania. Another doc I know works in the UK for three months and earns enough that she can spend the rest of the year working in Romania without having to ask for these 'tips'.
So, if she's using the 'NHS', then she would definitely need some money up front.
Just by searching online a little, a caesarian at a private clinic costs about 1000 pounds (or more for some extra services, like single room etc) and includes food, analysis, consultancy, vaccinations, and so on. They'll be other costs throughout the pregnancy, but I guess as you said she's heavily pregnant she's already managed those. The 'free' state healthcare service will probably be (again, from browsing Romanian online forums) about 500 pounds for the operation itself, but they'll be a lot of extras here and there, you'll have to buy some of the medication yourself, tip the nurses and so on. I remember one friend telling me how she'd had one of her kids on the state, the other private, and in the end it didn't really make much difference financially.
If it's the state situation, they'd never accept to be 'paid later' as this would mean most people would do a bunk and never pay as there's no legal backing to the 'fee'.