Is it permissible to travel for Ziyarat e.g to Najaf and Kerbala during ramadhan for less then 10 days in each place? E.g. 1 week in Najaf and 1 week in Kerbala (from the UK). And if it is permissible, would we have to miss the fasts and repay later on or can we fast whilst over there?

Yes it is permissible to travel any halal travel and stay for less than ten days. You cannot fast if you are staying less than ten days. You would just have to make up the fast after Ramadan. No need to pay anything since you are excused for breaking the fast due to travel.