In a statement on its website, EMR says customers who have tickets for today (Friday) can use them for today or tomorrow (Saturday).
The Met Office has issued a rare red weather warning for the south west and parts of the south east, as well as amber warnings for much of the rest of England, including Nottinghamshire.
High winds are expected to cause major disruption to services with some routes already seeing a reduced number of trains and journey times extended.
On its website, EMR states that one train per hour will run between Nottingham and London St Pancras and journey times will be significantly extended with trains running up to 90 minutes later than advertised.
There will be no trains departing London St Pancras towards Nottingham after 7.05pm this evening.
Additionally, the section of the route between London and Luton is covered by a red warning, meaning trains will be subject to short notice cancellation as wind speeds increase.
The service to and from London St Pancras may also be withdrawn at short notice with no alternative transport available.
Matt Stacey, head of stations at EMR said: “As the storm strength and potential impact becomes clearer, we are now strongly advising customers to not travel at all today.
"The strength of Storm Eunice will mean that on Friday there is likely to be widespread disruption to services across the country’s rail network.
"If customers absolutely need to travel we suggest they check our website for the latest information before setting off. They should also leave themselves plenty of extra time to reach their destination."
If you would rather not travel, and claim a refund instead, you can find information about how to do this here.