IN GB you canNOT REServe online-nor does one really need to. Its free when done at normal stations on that funny island in the northsea and no, you do not really need to do that many weeks or monthes advance. EXception are the very few sleepers in the 2 or 3 overnight trains (to Scotland and Cornwall)

Well-if you insist or get the grips without-it can be done online by creating some kind of account on 1 of the dozens of railways they have there and then from that account with some special moves.

INside FR there is a quota system for TGV-as has been told here a few zillion times now- 10€ if lucky, 20 if not-but the app canNOT handle the 20€-use other means-also explained here a few trillion times.

For TGV (=the superfast trains FRance is so world-renownd for) one also canNOT REServe for trips over the border and these can cost an awful lot extra. Nearly always there are cheaper ALTernatives by local=regional=free trains for that sector=means more changes and thus longer total trip time.

In case you follow that now fully trodden path anyone seems to want: it gets even much harder in ESpana-only locally and in hi-summer often completo=full days ahead.

Thank you for replying, although you did not really answer my question. Perhaps I was not clear.

Specifically, I can not book a reservation on a Eurostar train between London and Paris. I am receiving an error message that seems to be a problem with the Eurail website. All the information I have read says I should be able to easily book this train on the Eurail website, yet it does not work. 

I tried booking this train through b-europe.com instead. That website informed me that I need a pass cover number to complete the reservation. I do not have one from Eurail, since I have a mobile pass, not a paper pass. I sent a request to Eurail 2 days ago to request my pass cover number (as the FAQ instructed me to). I still have not received a response.

I am guessing that the Eurail website is only semi-functional. The support staff seems to be overwhelmed and is simply not able to help customers in a timely manner.

Does anyone have a recommendation on how to book a London to Paris Eurostar train reservation through Eurail? When will the Eurail website be fully functional? How long should I expect to wait to get a response from Eurail customer support? 

It might be that there were no more seats available on this particular trip. If you wish to reserve through SNCB International, then you should be able to generate a pass cover number by using this tool.