How to do London & Cardiff areas family of 4

It is regrettable that Air Transat no longer have direct flights between Toronto & Cardiff as with lower landing fees than London airports, they were able to offer good value fares. Anyway, I think you still check them out for flights to places in England.

On landing, I suggest you firstly head to Bath for a night or 2 as it is easy to walk around and the train station is right in the middle. (Obviously check the location of any hotel). From Bath, you can take a direct train to Cardiff and these trains are hourly and the journey takes just over an hour. https://visitbath.co.uk

Obviously, going to Cardiff does not mean that you have seen Wales - which deserves at least a week. If you are looking for good value accommodation in the central area of Cardiff check out Sleeperz Hotel & Premier Inn (both near Cardiff Central station). Another Premier Inn is found on Churchill Way near Queen Street station but this one might be prone to building work nearby for a tower block.

Loads to see in and around Cardiff including 4 very different castles. Pontypridd is a rather run down town in a former coal mining area. Just hop on one of the frequent (like every 15 minutes) local trains to Pontypridd. However, you may like to buy return tickets to Trehafod - which is on the Treherbert line & stop off at Pontypridd on the way back to Cardiff. I suggest this because a short walk from Trehafod station is Rhondda Mining Heritage Centre. Stagecoach Bus 132 stops right outside & also stops at Tongwynlais for Castell Coch (see below) so may be preferable to train. Be sure to tell bus driver where you wish to get off as you don’t know the route.

https://www.cardiffcastle.com/the-castle/ is found in the centre and dates from Roman times. In the middle is a Norman keep (1087) but much of the castle dates from the 1800’s when it was owned by the Marquess of Bute.

CAERPHILLY CASTLE is a massive medieval fortress just a 20 minute local train ride north from Cardiff. The town of Caerphilly is a dump in an ex mining area but the castle is everything that you would expect of an 1100’s fortress complete with moat. If you are visiting Caerphilly, you might like to break the journey at Lisvane station and walk for 5 minutes across the car park and into CEFN-ONN-PARK (free). In spring it is very colourful with azaleas, camellias, rhododendrons and bluebells all under the canopy of a temperate rain forest. https://www.outdoorcardiff.com/parks/parc-cefn-onn/
https://cadw.gov.wales/visit/places-to-visit/caerphilly-castle

CASTELL COCH stands on a hillside in north Cardiff. If visiting by bus from Cardiff - you need Stagecoach 132 from Greyfriars Road and alight at Tongwynlais (about 25 minutes out). The walk from the bus stop to castle is about 17 minutes uphill.
https://cadw.gov.wales/visit/places-to-visit/castell-coch

ST.FAGANS CASTLE & MUSEUM OF WELSH LIFE is just to the west of the city. (Cardiff Bus 32 or First Bus 320 - both not Sundays) The site is vast and needs at least half a day. You have grounds with buildings of significance brought from all over Wales and re-erected. The Castle is really a 1500’s Manor House. You will also find gardens near the castle. You will also find indoor galleries explaining about life in Wales through the ages - handy if it starts to rain. https://museum.wales/stfagans/about/

If visiting Cardiff, I suggest you visit CARDIFF BAY - which is just over 1 mile south of the city centre.
https://www.visitwales.com/destinations/south-wales/cardiff/top-things-see-and-do-cardiff-bay

THE NATIONAL MUSEUM OF WALES is found just N of the shopping area next to the City Hall. It has extensive art galleries.

https://www.visitcardiff.com
https://www.visitwales.com is the main official site for tourism to Wales.

James gave you great advice.
We did South Wales in 2017, renting a car and driving to Bath before heading to Wales. We loved Wales. My surname is Welsh and my ancestors left their village in SW Wales in 1716, We visited it in Cardiganshire and that was wonderful.
We spent two nights in Cardiff, then two nights in Tenby before moving on to St. Davids and the village of my ancestors, before heading back to England for more touring.

Driving from Bath to Cardiff, we enjoyed the scenic areas and visited an ancient abbey and castle. You can take the train, but you would miss seeing those types of places.
We stayed at a hotel in Cardiff that was downtown, so we could walk to where we need to go or take a train. However, parking was a bout a quarter of a mile from our hotel at a high rise parking garage.

Not sure of the rail connections from Cardiff to other places in S. Wales, but Tenby and especially St. Davids were wonderful.
St. Davids is located on the peninsula at the tip of SW Wales.

The Cotswolds are worth a visit, but a bit north of a direct route from Heathrow to Cardiff. Also, the rail connections to the Cotswolds are not good. You need a car to explore this area.

One very easy place you could go to between Bath and Cardiff by train is Cheltenham which tags itself as Cultural Centre of the Cotswolds. This is a good sized town (population 118,000) so lots of retail and food opportunities and is a Regency spa town.

From there there are several bus routes into the Cotswolds, enabling you to see them without a car. These routes include an hourly route through the Cotswolds to Moreton in the Marsh.

But also from Cheltenham Racecourse there is the Gloucestershire Warwickshire Steam Railway- this goes up to the well known village of Broadway (The 'Jewel of the Cotswolds').

The train also calls at Winchcombe for Sudeley Castle and Gardens.

Both Winchcombe and Broadway also have regular bus services.

Also between Cheltenham Spa and Cardiff on the train you have Chepstow- Castle, walls and Priory Church (with it's links to Oliver Cromwell and Anne Boleyn)-so plenty there for the historian.

As you only have 8 days there is no reason why you can't go direct from Heathrow to Cheltenham- either by train or direct by National Express bus every 90 minutes or so (journey time 2 hours 15 minutes).
Cardiff also has an hourly direct National Express bus from Heathrow- taking about 3 hours.

So you could do two days Cheltenham/Cotswolds (the first being arrival/jetlag day), two days Cardiff/Pontypridd and the balance in London. You could even squeeze Cardiff or Cheltenham to 3 days.

Thank you to all who replied. This is helpful information, very much appreciated!