I agree with Steve. The website is the right one for checking on trains. I've checked with www.tfl.gov.uk which is for all transport in London. It confirms that the route is step-free.
There are considerable engineering works that weekend on the Underground and railways but not on the route via Green Park.
There is a direct train from Marylebone Station to Shrewsbury at 12.24 but it is very slow, but pleasant through the countryside - and you don't have to change. Your big worry with that one would be it takes at least an hour to clear customs and immigration and you would be VERY tight on time. If you missed that Wrexham and Shropshire train it is 5 hours to the next one. Only £10 but you should be able to do that but Virgin can be 50 pence cheaper.
The Virgin from Euston runs every 20 minutes and you do have to change trains in Birmingham. (At Birmingham New Street, not Birmingham International). If you reserve now I saw many fares at £9.50. If you miss the train you reserved though, you will have a worthless ticket and have to buy a full priced one at £44.80, so you want to give yourself plenty of time. Or pay the full discounted off-peak fare of £44.80 and just hop on the first one when you get to Euston!
Shrewsbury to Bath is not simple. Shrewsbury to Birmingham, change for Bristol Temple Meads, change for Bath is the apparently fastest route at 3:15 or so, but if you take the very scenic route through Wales and change at Newport for Bath you will save a connection and half an hour.
Railways in the UK are decentralized and privatised so your journeys above are on Virgin, London Midland, Arriva Wales (now part of DB, the German railway), Cross Country, or First Great Western. They're all different. Vive le difference!
