Found this online:
A double-barrelled surname
If you and your partner want to take each other’s name as a double-barrelled surname and it’s clear how the name is derived, then your marriage certificate is sufficient evidence of your name change — you don’t need a deed poll.
Note that if it isn’t clear how the double-barrelled surname is derived, then the name will be considered a new surname, and you’ll need a deed poll as evidence of your name change.
So for example, if your surname is Jones and your partner’s name is Smith, then Jones-Smith and Smith Jones would both be clearly derived from your respective surnames, and a deed poll isn’t necessary to change your name. (It doesn’t matter what order you put the surnames or whether or not you use a hyphen.)
However the surname Cox-Jones-Smith isn’t clearly derived because it introduces an extra element.
Although double-barrelled surnames are becoming more popular, you may still have difficulties having your marriage certificate accepted by some independent organisations, such as banks, so it’s a good idea to renew your passport first. An updated passport is strong proof of your new name, and organisations are legally required to recognise your change of details under the GDPR.
If, on the other hand, you aren’t planning on updating your passport in the near future (it’s an extra cost of £ 82.50, and it’s not a legal requirement to have it updated), then you might consider changing your name by deed poll, to save yourself any possible hassle.