Is my European smartphone already unlocked?
Yes. Most likely.
The only commonplace form of locking that I am aware of that is used by vendors of phones in Europe is locking the handset to SIMs issued by a specific phone-network when that network subsidises the initial cost of the phone. When purchasing a phone you usually have a choice of monthly-tariff, pay-as-you-go (UK term) or SIM-free. The SIM-free option is always unlocked in my experience.
Note that there are also various "locks" that the end-user can enable on the phone hardware and/or separately in the SIM - basically a kind of password. These are not relevant.
I changed providers
When you change network providers in the EU (or at least in the UK), the phone must be in an unlocked state. If it was previously locked, you typically pay the existing provider an unlocking fee - after which the phone can be used with any provider's SIM.
You might not need to unlock the phone, if you change between providers that use the same underlying network, for example when there is one or more mobile virtual network operators (MVNOs) using the same underlying network. E.g. changing SIM from one provided by a network operator to an MVNO using their network or between two MVNOs using the same network.
I've never had any problem with a phone bought in the UK used in other parts of the planet. There can be problems with frequencies supported - in the early days you paid a premium for tri-band or quad-band GSM phones that would work in the USA (there were basically two types of phone, ones that worked worldwide except the USA and ones that worked in the USA). Nowadays the situation is far more complicated. I just wing it, I figure I can always buy a cheap phone at the other end if things don't work out.