When looking at old records be aware that most were written in Latin. Thus Richard would perhaps have been recorded as Richardus
For Richard Lovell to be tutor to Henry Stuart he must have been a well known and learned man. In that event I look in http://www.theclergydatabase.org.uk and there I found Richardus Lovell who was first of all Curate at Shorne between 1602 and 1605. He was ordained as Deacon on 15/11/1599, and Priest on 25/11/1599.
Now I think he may have been too old to be the tutor but if he had a first born son there is a high probability that he would also have been named after his father as was the custom of the time.
Next stop is Oxford University Alumni, 1500-1886 listing which is unfortunately held on Ancestry website so this becomes a cost. There is however one Richard Lovell listed as an Alumni which you may wish to follow up. Normally these records contain biographical information so may be worth a punt. I found a fathers name address and occupation for one of my look-ups which took me on to a new line of enquiry.
Cambridge University Alumni lists 6 Lovell entries but none are for Richard or Richardus. However you can search their data for free on http://venn.lib.cam.ac.uk/Documents/acad/enter.html.
Overall I guess that Ancestry looks like your next port of call.
Good luck with your researches