In reply to monkeydan:
There's nothing worthwhile without driving at least an hour.
The nearest place that I would recommend is probably Cademan Wood and Broad Hill in Leicestershire. Lots of small granite boulders and buttresses scattered around the woods. There's a link to the pdf guide on the crag page. Some of the boulders in the guide are a bit small and underwhelming but there's some good stuff there too. Use the GPS coordinates in the guide to find things or it is really easy to get lost despite the relatively small area.
In Leicestershire you also have:
Forest Rock which has the best bouldering in the area but you really need to be able to climb at least 7A to get much out of it so not a good place for a beginner.
Beacon Hill Some limited bouldering in a country park. Some people like it there, others complain about weird slippy rock and bad landings. I've never been (make of that what you will).
There's some other bits dotted around Leicestershire, some of which is worth a visit, much of it isn't.
If you drive a bit further, there's a lot more quality and variety in and around the Peak District. Pick up a Peak District bouldering guide and see what takes your fancy.
Be aware that the access situation at many crags is quite fluid at the moment due to COVID-19 restrictions. Lots of car parks are still closed. It's worth checking these forums and the crag page on UKC and the BMC's RAD database before visiting to check if there are any issues and make sure you park considerately wherever you go.
Of the stuff much closer to Milton Keynes/Bedford, people climb them out of desperation, not because they are any good.
Clifton Reynes bridges are best avoided. They are loose and dusty and don't expect a friendly welcome if the farmer sees you! The Robin Hood in Clifton Reynes is well worth a visit though if you can find it open. There are various other bridges, statues and other assorted choss around Milton Keynes, Kempston and Northamptonshire that have been climbed but they really aren't worth the effort.
As mentioned further up, Slawston Bridge in Leicestershire is the best bridge. We had an entertaining afternoon climbing on it once on the way back home after getting rained off at another crag. But it is a long way to go to climb at a bridge when a little more driving can take you to some fantastic places.
I've heard there is some rock beyond the main kids play area in Emberton park. I'm not aware of anyone who has ever climbed it and am not sure if it is the artificial boulders mentioned above or something else. There is a couple of artificial boulders at Stanwick lakes but once you've driven that far, you might as well keep going on to somewhere decent.
I work a couple of miles from Stanwick and my parents live within walking distance of Clifton Reynes and Emberton. I have still never bothered climbing at Stanwick or Emberton and haven't climbed at Clifton Reynes for 8 years.