There are no announcements for Chiltern Railways services at the Metropolitan Line stations (Harrow On The Hill / Rickmansworth / Chorleywood / Chalfont & Latimer / Amersham). Unfortunately the automated announcements system completely ignores these trains. There are not even any recording for "Chiltern Railways" or "Aylesbury" or "Aylesbury Vale Parkway" or "London Marylebone" etc on their Telent automated announcements system. It is ridiculous as TFL could easily get Elinor Hamilton to do these additional recordings so Chiltern Railways services can be announced. It seems that TFL do not really care as Chiltern Railways are nothing to do with them. I hope they update the system and add these trains eventually.
Unless something has changed relatively recently, I believe there are no voice announcements (to announce next or approaching trains) at all north of Harrow on the Hill. In fact, I think the first station going into London where trains are announced may actually be Wembley Park. Therefore on the section you reference which are shared with Chiltern, I am sure there are no announcements of Met Line trains either. The visual displays on the Northbound do tell you the destination of the train though. but going South, Chiltern trains are displayed as "Chiltern Railways"
It is rather odd how LU do not make much effort with National rail over their section yet vice versa where the Bakerloo and District run over NR tracks, a full effort is made to integrate LU services into announcements. However, whilst I agree that LU are nowhere near as good as NR with announcements, we must also remember that the Bakerloo and District run over LO territory, which is run by TFL from a visual perspective at least and it would be highly unlikely that announcements would not provide for all services. This means that the Met line is the only line that actually integrates with a "real"** National Rail service where the rolling stock used isn't even electric.
What's my point about whether its "real" NR or LO?.......Well, prior to LO taking over what was ex Silverlink, I do not believe there were any announcements at all for LU or Silverlink trains on the shared sections with the District or Bakerloo (although if there were, please do correct me). It could therefore be said that NR would also not be that bothered about anything which was TFL run****
I have used the section between Moor Park and Harrow my entire life and have never once heard any announcement to do with the next or approaching trains.
**Yes, I know LO is technically an NR service....
****I am aware that more recently, stations such as Richmond announce District line services (although only the destination). Not sure if there are any other examples where a NR station that is non LO owned station does this though.