are british trains on time

Eurostar trains require you to check in "at least 30 minutes before departure, 45 minutes at busy times like weekends and bank holidays." Allow another 15 minutes to navigate the station and 15-30 minutes for the taxi. So I'd want at least two hours between trains.

Given that Eurostar requires check-in at least 30 minutes before departure (and earlier at busy times), then , no, 45 minutes will not be even close to sufficient. Rome2rio.com gives this as what appears to be an unrealistic 7 minutes (for 3 miles), Google maps 11 minutes, without traffic. Neither accounting for time on foot from train to street and street to check-in. Our Londoners will be able to tell you if it is possible to have a cab reserved and waiting at Paddington. Do you really wish to risk missing your train by cutting it close? (Edit- Laura's reply showed while I was typing this)

British trains are normally on time. However, when things go wrong, they can throw everything off schedule. You can see if the trains are running on time today at http://www.realtimetrains.co.uk/search/advanced

Now, I have not tried it myself but I understand that if you book a ticket for the complete journey via the following website, that in the event of failing to get the ‘correct’ Eurostar due to earlier delay (from Bath), that you have the right to use the next Eurostar - space permitting. I have not checked the fares on this site compared to paying separately for the different legs of the journey.https://loco2.com

You may also like to know that you could fly direct to Paris from Bristol Airport - which is not very far from Bath.
www.easyjet.com
https://www.bmiregional.com/book/book-a-flight/

Buy a through ticket from Eurostar - these may now only be available on the UK Eurostar website. This gives you a ticket under the 'CIV' regulations which means that should your first train be delayed, Eurostar is obligated to put you on the next available service should you arrive too late for the one you are booked on. Transfer included via the Hammersmith and City Line, which is now step free at both ends.

Loco2 sell tickets to London Int'l CIV - this is usually more expensive than with Eurostar but not always so price up both.

If you are unable to get tickets Bath to Paris direct from Eurostar, then ensure your Bath-London tickets are to London CIV and not just London Terminals (see seat61 http://www.seat61.com/UKconnections.htm#.V7a2yhnTWBY). If you arrive late in Paddington and get the "CIV" ticket endorsed by station staff, you're protected on later Eurostar services, as I understand it.

I can confirm what James said, If you book both tickets as one transaction they count as a through journey, and if you miss the second train because the first train is late they have to accomodate you on the next available train.
I use https://loco2.com and if you book it as one transaction through them this rule applies, It happened to me once (delays due to snowstorm), ask the conductor to stamp your ticket and there will be no problem.

Trains are usually on time (within 5 minutes 90-95% of the time. But if they go bad it can be very bad, 20-40 minutes late.
I would allow 30 minutes minimum between stations, plus Eurostar's check-in time + safety = about 2 hours.

This is easy enough to do by tube (Circle/hammersmith line), and probably quicker than taxi. But it may be difficult if you have haevy luggage as at Paddington the Circle/Hammersmith line is on the surface in the main station, so you have to go UP 2 flights of stairs and across a bridge to get to it. At St Pancras it is all escalators or lifts.

There are no stairs required to navigate to get to the Hammersmith and City Line platforms as there is a slope up and a lift down. The stairs might be a shorter route if you are in the country end of the train but there is no need to use them.

Around 90% of British trains arrive within 5 minutes of scheduled time. That won't help if it is your train in the difficult column on the day.

Since you need to be checked in 45 minutes before departure, I prefer longer myself - like one hour - there is no way that arriving at Paddington 45 minutes before check in will work unless you can apparate. If so, you will not need the train at all. Otherwise you will be out of luck.

So this must be what that taxi is about in your other post. Nope, won't get you there in time. Better to reschedule, especially when coming from so far away, and be sure to use CIV, and travel the tube for no more money.

This is really good information. I did not know about the CIV tickets or the loco2 website. Thanks to all.

You can't arrange taxis ahead of time--there is a taxi rank near platform 12 at Paddington where taxis line up.

You can however arrange for a private hire car (minicab) to meet you. Not recommended by me for this transfer though.

Seat 61 gives excellent instructions on the CIV provisions and exactly what you need to click on on the Eurostar site to make that option appear.

There have been some, rather irritating, changes to the availability of CIV tickets: http://www.seat61.com/news.htm#.V8wIkOvTVSA

You must now either buy them at the same time as your Eurostar tickets, or purchase them in person at a station (presenting your Eurostar ticket as proof you're not trying to scam the system).