Have commuted Bristol to Chippers by bike a few times. A4 and A420 both fast and deadly in places and A4 takes you a long way out. Agree with AnalogueAndy that regional route 17 is a good bet, but don’t join it directly from A420 and Castle Combe as you’ll go down a hill and back up it (unless you want the practice!)
So, head out of Chippenham on A420 but 2km past ring road (A350) turn right on to B4039. Follow through Yatton Keynell, UPPER Castle Combe (Castle Combe proper is down that hill), The Gibb and Burton. Coming out of Burton, rather than following the A4039 uphill, turn off left on a minor road. This road parallels the motorway, past two roads on the left to a crossroads where it meets regional route 17. Straight over crossroads and on another 2 km to A46. Cross A46 (busy), carry on and down a steep hill to Hinton. Straight on, up a hill and turn right at a T-junction, bringing you into Pucklechurch. Abandon regional route 17 here and instead follow B4465 south west out of village. After a sharp right bend and after another kilometre look for a turning on the left signposted Shortwood, just before a hill down. Take this turn and follow steeply downhill through traffic calming, then levels out and crosses Bristol ring road (A4174). IMMEDIATELY after crossing ring road, at end of bridge, turn right at traffic calming island onto cycle path. This descends and loops round to pass underneath the bridge you’ve just been over. Follow path along side of ring road for a bit and it gradually peels away from the ring road and after several hundred metres meets the Bristol to Bath Cycle Path at a disused station. Turn right and follow the path all the way to the centre of Bristol (gets busier the nearer you get)
I’ll stop there and see if anyone else can give good directions from the end of the cycle path to the Old Vic. I could give you some directions but I don’t do it in rush hour so don’t know what’s best regarding traffic/safety.
Hope you can follow all this. Do you have an OS map or mapping software?