Nobody move or the mother and baby get it!

From The Royal College of Midwives:

As a union the RCM argues that being part of the EU has led to better working conditions, and will ensure these remain in place. These benefits include equal pay for women, a guarantee that pregnant women get paid time off to attend antenatal appointments, minimum holiday entitlements, maternity rights and protection for those working part-time. These are rights worth protecting, and so long as the UK remains in the EU they are rights that UK Government’s [sic] cannot take away.

Remaining in the EU also helps to maintain safety in the NHS and promotes high standards among healthcare professionals. There are over 33,000 nurses and midwives from other EU countries working in the NHS. EU rules mean that they must have training and skills equal to UK trained staff. This helps to ensure high standards and good quality care in our NHS.

Healthcare colleagues from across the EU also help to fill the vacancies on our maternity units and wards.

Membership of the EU helps the UK Government to have the money to invest in the NHS and other public services. It is the taxes that flow from a growing economy that pay for public services like the NHS. The economic turbulence that would follow a UK exit from the EU would jeopardise this.

The Equal Pay Act was first enacted in... 1970.

Do they seriously think that a future UK government would reverse the right to time off for ante-natal care?

Let's accept that staff training standards are high in other EU member states and so they are more or less automatically qualified to work in UK hospitals. What changes after Brexit..?

Yes, the NHS is disproportionately staffed by people from abroad at all skill levels, many are from other EU member states, many are (poached from) from non-EU member states i.e. developing countries. No sane politician is going to put a stop to this, the NHS would grind to a halt overnight.

Why do they think that "economic turmoil" would automatically follow Brexit?

Midwife Shortages

From the BBC 11th October 2015

The Royal College of Midwives (RCM) said a shortage of midwives was having a major impact and mistakes would "almost certainly be made".

From The Guardian August 18 2013

The calculations, by the Royal College of Midwives (RCM), suggest that the gap between the number of midwives the NHS in England needs and the number it now has will not be closed until 2026. The shortage is certain to force maternity units to close suddenly and lead to some mothers receiving inadequate care before, during or after giving birth.

From the Daily Mail March 18 2012

Mary Newburn, head of policy research at the NCT, said: 'There is a major retention problem across the UK but moving staffing provision from midwife-led units into hospital delivery suites is not the answer, and it is certainly not what women want.
'When community-based, midwife-led units are closed, women often have to travel miles to hospital to have their babies delivered by unknown, stressed staff, which can lead to tension and further complications, including increased intervention.'

From BBC 15th September 2011

Parts of England are facing big midwife shortages, putting mothers and babies at risk, midwives have said.