I really do not think it is necessary to fly to Faro.
Ryanair fly Birmingham to Porto and are showing flights twice a week in December on Mondays and Fridays.
Flight No.FR 1072 16:15 Depart Brums 18:45 Arrive Porto.
From Porto Airport you can get a metro to Porto Campanha taking about 30 minutes and then a train from Porto to Coimbra.
You could then catch the 20:47 train from Porto Campanha arriving Coimbra-B at 21:44.
Alternatively, Easyjet fly several times a week from Luton to Lisbon. Prices for December as very low but departure time is very earlier at circa 06:35 or 08:35.
Again, fly to Lisbon, grab a cab for € 10 to Gare Oriente Station. There are hourly trains to Coimbra taking between 1 hour 45 min and 2 hours.
To see the timetable for the route go to my blog at
http://barlaventostravelblog.blogspot.com/
Algarve & Portugal - Travel Transport Links
You will find a link entitled:
"Porto - Lisbon - Porto Train Timetable"
Click on it and you will be able to download the current timetable from Portuguese Railways for the Lisbon - Coimbra - Porto service.
Below you will find the timetable for the Faro Line as well.
Hope this helps.
