'2017'

Local NHS trust is a client of mine, so had it bookmarked…

At least this is what it ought to be…
https://www.england.nhs.uk/nhsidentity/identity-guidelines/fonts/

This (with some modification)

"TEACHER" & "WERE MEANT TO BE"

for a cookie, the border looks to be made using this…

everything else… (looks like the maker created a stencil from the 2 fonts)

'Dragons' is based on this

This is hand lettered. The LoC stamp in the bottom right says 1894.

'KINGS' with custom fill

I think this is hand lettering for a couple of reasons 1. It dates back to 1981 and 2. there are inconsistencies in the letterforms. See overlay here

'CHRISTMAS'. It has been scaled…

Hasn't changed much since 1916, so the work of a sign painter, rather than a typeface. There may be a digital homage/revival, but not that I'm aware of, and the identity of the original designer is apparently a mystery
http://www.printmag.com/logo-design/nathans-famous-100-year-old-logo/